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Microlocal methods for dynamical systems
QMath 13 Georgia Tech
Maciej Zworski
UC Berkeley
October 10 2016
Dynamical systems
Dynamical systems
Dynamical systems a statistical approach
Dynamical systems a statistical approach
Linear Non-linear
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical systems
Dynamical systems
Dynamical systems a statistical approach
Dynamical systems a statistical approach
Linear Non-linear
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical systems
Dynamical systems a statistical approach
Dynamical systems a statistical approach
Linear Non-linear
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical systems a statistical approach
Dynamical systems a statistical approach
Linear Non-linear
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical systems a statistical approach
Linear Non-linear
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical systems a statistical approach
Completely integrable Chaotic
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
In the chaotic case positions and directions get uniformlydistributed
Typical questionsHow long do we have to wait to have uniform distribution
Are there periodic orbits and what information to they contain
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
In the chaotic case positions and directions get uniformlydistributed
Recent work on the rate of decay for billiards byBaladindashDemersndashLiverani rsquo15
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function
Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
Replace p by log `γ where `γ is the length of a prime closed orbit
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A dynamical analogue of the Riemann zeta function Ruelle zetafunction
Replace primes with prime closed orbits in ζ(s) =prod
p(1minus pminuss)minus1
ζD(s) =prodγ
(1minus eminuss`γ )
Replace p by e`γ where `γ is the length of a prime closed orbit
It turns out that the zeros and poles of ζD contain informationabout statistical properties of the chaotic dynamical system
That includes the time at which we achieve uniform distribution
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Y (`1 `2 φ) for `1 = `2 = 7 φ = π2
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC
ldquoI must admit a positive answer would be a little shockingrdquo
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Dynamical zeta functions have been studied by many authors
Selberg rsquo56 ArtinndashMazur rsquo65 Smale rsquo67 BowenndashLanford rsquo68Ruelle rsquo76 MilnorndashThurston rsquo77 ParryndashPollicott rsquo83rsquo90 Pollicottrsquo86 CvitanovicndashEckhardt rsquo91 Mayer rsquo91 Rugh rsquo96 Fried rsquo86 rsquo95Kitaev rsquo99 PetkovndashStoyanov rsquo07 BaladindashTsujii rsquo08 Stoyanov rsquo11FaurendashTsujii rsquo13 BorthwickndashWeich rsquo15
Smale rsquo67 conjectured that for Anosov flows ζD is meromorphic inC ldquoI must admit a positive answer would be a little shockingrdquo
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
What is an Anosov flow
TρX = E0(ρ)oplus Es(ρ)oplus Eu(ρ) ρ 7rarr Ebull(ρ) continuous
dϕt(ρ)Ebull(ρ) = Ebull(ϕt(ρ))
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Eu(ρ) t lt 0
|dϕt(ρ)v |ϕt(ρ) le Ceminusθ|t||v |ρ v isin Es(ρ) t gt 0
()t
+
E
E
Example X = SlowastM = (x ξ) isin T lowastM |ξ|2g = 1 where (M g) isa compact Riemannian manifold of negative curvature
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Theorem (GiuliettindashLiveranindashPollicott rsquo12 DyatlovndashZ rsquo13)
For an Anosov flow on a compact manifold ζD(s) has ameromorphic continuation to C
Our proof uses the microlocal approach to Anosov flows due toFaurendashSjostrand rsquo11 and radial point propagation of singularitiesestimates due to Melrose rsquo94 and developed further by Vasy rsquo13
The noncompact case (essentially the full Smale conjecture)recently completed by DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
x
ξ
ξ2 + V (x) = E
T lowastRn
x
ξ T lowastMξ(Xx )
=0
Scattering theory FaurendashSjostrand
0
Q
partTlowastM
Elowastu
Elowasts
DyatlovndashZ DyatlovndashGuillarmou
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Special case
Let (M g) be a smooth oriented compact Riemannian surface of
negative curvature
ζD(s) =prodγ
(1minus eminuss`γ ) Re s 1
Theorem (DyatlovndashZ rsquo16)
Let g be the genus of M Then s2minus2gζD(s) is holomorphic nears = 0 and
s2minus2gζD(s)|s=0 6= 0
In particular the set of lengths of closed orbits (the lengthspectrum) determines the genus
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X )
eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Pollicott-Ruelle Resonances
Correlations f g isin Cinfin(X ) eminusitP f (x) = f (ϕminust(x))
ρf g (t) =
intX
eminusitP f (x)g(x)dx
Power spectrum
ρf g (λ) =
int infin0
ρf g (t)e iλtdt
Resonances poles of ρf g (λ)
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A ldquorealrdquo life example
Rough parameter dependence in climate models and the role ofRuellendashPollicott resonancesChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
ρf g (t) =
intX
eminusitP f (x)g(x)dx
PollicottndashRuelle resonances are the poles of ρf g (λ)
m(λ) = dim
u isin Dprime(X ) (1i V minus λ)ru = 0 WF(u) sub E lowastu
E lowastu = (Eu otimes E0)perp sub T lowastX
BlankndashKellerndashLiveranirsquo02 FaurendashSjostrand rsquo11
To obtain exponential decay of correlations one needs a gap
ν0 = supν no resonances for Imλ gt minusν
ν1 = supν finite n of resonances for Imλ gt minusν
ν1 gt 0 for contact flows (and in particular for geodesic Anosovflows) Dolgopyatrsquo98 Liverani rsquo04 Tsujiirsquo12 NonnenmacherndashZrsquo15
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
A ldquorealrdquo life investigation of the gap
Rough parameter dependence of the spectral gap in climatemodels ChekrounndashNeelinndashKondrashovndashMcWilliamsndashGhil 2014
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Microlocal analysis (semiclassical version)
I Phase space (x ξ) isin T lowastX
I Semiclassical parameter hrarr 0 the effective wavelength
I Classical observables a(x ξ) isin Cinfin(T lowastX )
I Quantization Oph(a) = a(x hi partx
) Cinfin(X )rarr Cinfin(X )
semiclassical pseudodifferential operator
Basic examples
I a(x ξ) = xj =rArr Oph(a) = xj multiplication operator
I a(x ξ) = ξj =rArr Oph(a) = hi partxj
Classical-quantum correspondence
I [Oph(a)Oph(b)] = hi Oph(a b) +O(h2)
I a b = partξa middot partxb minus partxa middot partξb = Hab etHa Hamiltonian flow
I Example [Oph(ξk)Oph(xj)] = hi δjk
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u
Oph(b)u+ hminus1
f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Standard semiclassical estimates
General question
P = Oph(p) Pu = f =rArr u f
Control u microlocally
Oph(a)u Oph(b)u+ hminus1f +O(hinfin)u
Elliptic estimate
p = 0 supp a
Propagation of singularities
supp ab 6= 0etHp
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Phase space description of singularities
u isin Dprime(X ) 7minusrarr WF(u) sub T lowastX 0
(x ξ) isinWF(u)
mexist a isin Cinfinc (T lowastX ) a(x ξ) 6= 0 a(x hD)uL2 = O(hinfin)
Examples
I WF(δ0) = (0 ξ) ξ isin Rn 0
I WF((x minus i0)minus1) = (0 ξ) ξ gt 0
I WF(δ(ax minus y)) = (x ax minusaη η) y isin R ξ isin R 0
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ )
=ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Pγ is the linearized Poincare map
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Return to the Ruelle zeta function (with a slight change ofconvention)
ζD(λ) =prodγ
(1minus e iλ`γ ) =ζ2mminus1(λ) middot middot middot ζ1(λ)
ζ2m(λ) middot middot middot ζ0(λ) 2m + 1 = dim X
ζk(λ) =
(minusinfinsum
m=1
sumγ
e imλ`γ trandkPmγ
m| det(I minus Pmγ )|
) Imλ 1
Theorem (DyatlovndashZ rsquo16)
ζk(λ) extends to an entire function and the multiplicities of itszeros are given by
dimu isin Dprime(X Ωk
0) (1i LV minus λ)ru = 0 WF(u) sub E lowastu
where Ωk
0 are k-forms satisfying ιVu = 0
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)
Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
The starting point for relating the zeta function to thedistributional spectrum of 1
i LV is is the AtiyahndashBottndashGuillemintrace formula
tr eminusitPh =infinsum
m=1
sumγ
`γδ(t minusm`γ)
|I minus Pmγ |
P = hV i
This is related to the first building block of the zeta function
ζ0(λ) = exp
(minusinfinsum
m=1
sumγ
e imλ`γ
m| det(I minus Pmγ )|
)Since
(P minus z)minus1 =i
h
int infin0
eminusitPhdt
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0
= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R
KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Here the trace is a formal trace obtained by integration over thediagonal
Au(x) =
intX
KA(x y)u(y)dy
tr A =
intX
KA(x x)dx
Allowed under a wave front set condition
WF(KA) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Example Au(x) = u(ax) x isin R KA(x y) = δ(ax minus y)
tr A =
intRδ((aminus 1)x) =
1
|aminus 1| a 6= 1
WF(KA) = (x ax minusaη η) η 6= 0
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
e it0λh trϕlowastminust0(P minus hλ)minus1 =part
partλlog ζ0(λ)
Need
WF(Kϕlowastminust0
(Pminusz)minus1) cap (x x ξminusξ) x isin X ξ isin T lowastx X 0= empty
Propagation through radial sinks (E lowastu ) and sources (E lowasts ) based onearlier work of Melrose rsquo94 and Vasy rsquo13 gives this
0
Q
partTlowastM
Elowastu
Elowasts
The meromorphy of z 7rarr (P minus z)minus1 Cinfin(X )rarr Dprime(X ) shows thatthe poles are simple and residues are positive integers
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Now suppose that X = SlowastM M an orientable Riemannian surfaceof negative curvature
What is the order of vanishing m of ζD(λ) at λ = 0
When M = ΓH2 (constant curvature)
Selberg trace formula =rArr m = 2g minus 2
In general
ζD(λ) =ζ1(λ)
ζ0(λ)ζ2(λ)
m = m1(0)minusm0(0)minusm2(0)
mj(0) = dimu isin Dprime(X Ωj
0) LrVu = 0 WF(u) sub E lowastu
where Ωj
0(X ) are j-forms satisfying ιVu = 0
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
Claim m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Regularity result
Re〈Vu u〉L2 ge 0 Vu isin Cinfin WF(u) sub E lowastu =rArr u isin Cinfin
Here L2(X ) is defined using the smooth invariant measure on SlowastM
dvol = α and dα
where α is the contact form
α = zdζ|SlowastM SlowastM = (z ζ) isin T lowastM |ζ|2g(z) = 1
α(V ) = 1 ιV dα = 0 kerα(x) = Eu(x)oplus Es(x)
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof
If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X )
Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)
Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = dimu isin Dprime(X ) V ru = 0WF(u) sub E lowastu = 1
Proof If Vu = 0 WF(u) sub E lowastu then by the regularity resultu isin Cinfin(X ) Also u(etV x) = u(x)Hence
〈du(x) v〉 = 〈du(etV x) detV (x)v〉 minusrarr 0
t rarr +infin v isin Es(x)t rarr minusinfin v isin Eu(x)
It follows that du|EuoplusEs = 0 and that means that du = ϕα
0 = α and d(du) = ϕα and dα =rArr ϕ = 0
=rArr du = 0
=rArr u = const
If V 2u = 0 WF(u) sub E lowastu then Vu = const
ButintX Vudvol = 0 so const = 0
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m1(0) = dim Y1 = dim H1(X C)
Y1 = u isin Dprime(Ω1(X )) LrVu = 0 ιVu = 0 WF(u) sub E lowastu
1 LVu = 0 is equivalent to (since ιVu = 0) to ιV du = 0 Hencedu is a resonant state for 2-forms and du = cdα Sinceu and dα = ιVuα and dα = 0 we have
cvol(X ) =
intdu and α =
intu and dα = 0
We conclude that du = 0
2 Hodge theory existϕ isin Dprime WF(ϕ) sub E lowastu uminus dϕ isin Cinfin(Ω1(X ))
Y1 3 u 7rarr [uminus dϕ] isin H1(X C)
is an isomorphism
Showing semisimplity is a little bit tricky
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention
m0(0) = m2(0) = 1 m1(0) = dim H1(X C)
Since for surfaces of genus g ge 2
H1(SlowastMC) H1(MC)
it follows that the order of vanishing of ζD at 0 is
m = minusm0(0)+m1(0)minusm2(0) = minusEuler characteristic of M = 2gminus2
Thanks for your attention