Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
The Weil-Petersson geodesic flow is ergodic
Amie Wilkinson(with Keith Burns and Howard Masur)
University of Chicago
INdAM Lectures
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 1 / 32
Teichmuller Space
Let S = Sg ,n = surface of genus g , with n punctures. (3g − 3 + n > 0)
Define Teichmuller space T = Tg ,n by:
T = { marked conformal structures on S}/conformal equivalence
= { marked hyperbolic structures on S}/isometry
Marked means each curve in S “has a name.”
Formally, an element of T is represented by a pair (X , f ), whereX = Riemann surface, and
f : S → X
is a marking homeomorphism. Equivalent definition:
T = {discrete, faithful rep’n ρ : π1(S) → PSL(2,R)}/conjugacy
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 2 / 32
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 3 / 32
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 4 / 32
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
0 1
z
T ∼= H
Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 3 / 13Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 5 / 32
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
0 1
same structure, different marking!
T ∼= H
Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 3 / 13Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 6 / 32
Properties of THomeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).Complex structure is natural, but not obvious.“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
0 1
H
different
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 7 / 32
Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼= H; σ = {α}
�α(x + yi) � 1
y, τα(x + yi) � x
yas y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 8 / 32
Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :Fenchel-Nielsen coordinates on T : “angle/action”Let σ = maximal collection of disjoint simple closed curves on S :
α1
α2
α3
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 4 / 13
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼= H; σ = {α}
�α(x + yi) � 1
y, τα(x + yi) � x
yas y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 9 / 32
Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼= H; σ = {α}
�α(x + yi) � 1
y, τα(x + yi) � x
yas y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 10 / 32
Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼= H; σ = {α}
�α(x + yi) � 1
y, τα(x + yi) � x
yas y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 11 / 32
Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :
Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :
�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼= H; σ = {α}
�α(x + yi) � 1
y, τα(x + yi) � x
yas y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 12 / 32
The Weil-Petersson metric
Theorem: [Wolpert] ∃ symplectic form ω on T s.t. for any maximalcollection σ:
ω =�
α∈σd�α ∧ dτα
Together with the almost complex structure J, this determines a (Kahler)Riemannian metric:
gWP(v ,w) = ω(v , Jw), (for v ,w ∈ TXT )
called the Weil-Petersson metric.
Example: once-punctured torus: T ∼= H, standard complex structure:
ω = d�α ∧ dτα � dx ∧ dy
y3; g2
WP � dx2 + dy2
y3, as y → ∞.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 13 / 32
Moduli space (where the curves have no name...)
Define moduli space M = Mg ,n by:
M = {conformal structures on S}/conformal equivalence
= {hyperbolic structures on S}/isometry
Mapping class group MCG = MCGg ,n = Diff+(S)/Diff0(S)
MCG acts (virtually) freely on T and is the (orbifold) fundamental groupof M:
M = T /MCG
In punctured torus example, MCG = SL(2,Z) and M = punctured spherewith 2 cone points.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 14 / 32
Properties of the WP metricIncomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞nor are they bounded away from 0, except in sporadic cases.(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X ,Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.Almost every geodesic on M is defined for all time and is recurrent.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 15 / 32
Properties of the WP metricIncomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞nor are they bounded away from 0, except in sporadic cases.(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X ,Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.Almost every geodesic on M is defined for all time and is recurrent.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 16 / 32
Properties of the WP metricIncomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞nor are they bounded away from 0, except in sporadic cases.(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X ,Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.Almost every geodesic on M is defined for all time and is recurrent.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 17 / 32
Properties of the WP metricIncomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞nor are they bounded away from 0, except in sporadic cases.(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X ,Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.Almost every geodesic on M is defined for all time and is recurrent.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 18 / 32
Properties of the WP metricIncomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞nor are they bounded away from 0, except in sporadic cases.(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X ,Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.Almost every geodesic on M is defined for all time and is recurrent.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 19 / 32
The WP Metric and deformation of hyperbolic structuresThe WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:
Limit set for fuchsian (hyperbolic) punctured torus
The WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:
Figure: Limit set for Fuchsian (hyperbolic) punctured torus
Theorem (McMullen):
d2
dt2dimH(Λt)|t=0 =
1
3
�X0�2WP
area(X0).
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 20 / 32
The WP Metric and deformation of hyperbolic structuresThe WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:
Limit set for quasifuchsian (projective) punctured torus (McMullen)
Theorem (McMullen):
d2
dt2dimH(Λt)|t=0 =
1
3
�X0�2WP
area(X0).
The WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:
Figure: Limit set for Fuchsian (hyperbolic) punctured torus
Theorem (McMullen):
d2
dt2dimH(Λt)|t=0 =
1
3
�X0�2WP
area(X0).
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 20 / 32
The geodesic flow
M = Riemannian manifold. T 1M = unit tangent bundle to M. Naturalflow ϕt : T 1M → T 1M called the geodesic flow:
Starting point: the geodesic flow
S = closed surface, Riemannian metric. T 1S = unit tangent bundle to S .Natural flow ϕt : T 1S → T 1S called the geodesic flow:
S
v
ϕt(v)
distance t
Basic question: is there a dense geodesic on S?
Stronger: is there a dense ϕt -orbit in T 1S?
Stronger yet: is almost every orbit (w.r.t. volume) equidistributed in T 1S?
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 3 / 19
Basic question: is there a dense geodesic on M?
Stronger: is there a dense ϕt-orbit in T 1M?
Stronger yet: is almost every orbit (w.r.t. volume) equidistributed inT 1M?
E. Hopf (1939): Yes, in the special case where M is a closed, negativelycurved surface.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 21 / 32
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the directionof the geodesic flow:
The geodesic flow in negative curvatureNegative curvature on S creates hyperbolicity orthogonal to the directionof the geodesic flow:
S
v
ϕt(v)
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 4 / 19
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 22 / 32
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the directionof the geodesic flow:
The geodesic flow in negative curvatureNegative curvature on S creates hyperbolicity orthogonal to the directionof the geodesic flow:
S
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 4 / 19
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 23 / 32
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the directionof the geodesic flow:
The geodesic flow in negative curvatureNegative curvature on S creates hyperbolicity orthogonal to the directionof the geodesic flow:
S
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 4 / 19
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 24 / 32
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the directionof the geodesic flow:
The geodesic flow in negative curvatureNegative curvature on S creates hyperbolicity orthogonal to the directionof the geodesic flow:
vϕt(v)
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 4 / 19
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 25 / 32
Hyperbolicity
If M is compact and negatively curved, then T (T 1M) = Eu ⊕ E c ⊕ E s :HyperbolicityIf S is negatively curved, then T (T 1S) = Eu ⊕ E c ⊕ E s :
vϕt(v)
E u
E c
E s
Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 5 / 19
where E c is tangent to ϕt orbits, Eu is expanded under Dϕt , and E s iscontracted under Dϕt .
(Hadamard) The geodesic flow is hyperbolic. Eu = unstable bundle, E s =stable bundle. Eu is tangent to unstable foliation Wu, and E s is tangentto stable foliation Ws . E c is tangent to orbit foliation O
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 26 / 32
The Hopf argument for ergodicity
Theorem (E. Hopf): If M is a closed negatively curved surface, then thegeodesic flow is ergodic (with respect to Liouville volume): for almostevery v ∈ T 1M, the ϕt-orbit of v is equidistributed in T 1M:
1
T
� T
0h(ϕt(v)) dt
T→±∞−→ 1
Vol(T 1M)
�
T 1Sh dVol, ∀h ∈ C 0(T 1S).
Idea of proof (“Hopf Argument”): Fix h ∈ C 0(T 1M). Birkhoff/vonNeumann Ergodic Theorems imply that for a.e. v :
h±(v) := limT→±∞
1
T
� T
0h(ϕt(v)) dt
exist and are equal. Moreover, h+ is constant along Ws leaves, h− isconstant along Wu leaves, and h± are constant along ϕt orbits (O leaves).
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 27 / 32
The Hopf argument, continued
h− is constant along Wu manifolds.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
A typical Wu manifold.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h− is const. along O manifolds (orbits of ϕ).
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h− is const along the O leaf through a.e. point.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h− is const along the O saturate of a typical Wu manifold.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h− is const along the O saturate of a typical Wu manifold.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
On this surface, h− is constant, and h− = h+ a.e.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h+ is constant along Ws manifolds.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h+ is constant along Ws manifolds.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h+ is constant along the Ws -saturate of this surface.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
h+ is constant in a box ⇒ locally constant ⇒ constant.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
The Hopf argument, continued
Hence ϕ is ergodic.
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 28 / 32
Ergodicity of geodesic flows
Anosov (1960’s): M = closed, negatively curved manifold, any dimension⇒ geodesic flow is ergodic (new ingredient: absolute continuity).
Question of when ergodicity holds is still open for:
• Closed, nonpositively curved surfaces.
• Complete, negatively curved surfaces of finite volume.
• Incomplete, negatively curved surfaces.
As a very special case of the latter, consider the WP geodesic flow onT 1T /MCG: is it ergodic?
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 29 / 32
Main Result
Theorem: [Burns, Masur, W] For any (g , n), the WP geodesic flow onT 1T /MCG is ergodic and has finite metric entropy. Almost every stablemanifold (horosphere) in T 1T is smooth and large.
Previous results on WP geodesic flow:
Transitivity: there exist dense geodesics [Brock-Masur-Minsky, alsoPollicott-Weiss-Wolpert for (1, 1) case].
Infinite topological entropy: there exist compact invariant sets withunboundedly large entropy [BMM].
Ergodic closing lemma: periodic measures are dense among ergodicprobability measures [Hamenstadt] .
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 30 / 32
The proof
The proof has three main ingredients:
Teichmuller theory
Differential geometry
Smooth ergodic theory
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 31 / 32
Input from Teichmuller theory:
• Asymptotics for WP metric and covariant derivative: Wolpertgives precise asymptotics to order 2 in special coordinates near ∂T .
• Asymptotics for higher derivatives of WP metric: (after McMullen).∃ totally real embedding of T into complex manifold (quasifuchsianspace) where ω extends to a holomorphic form with bounded primitive.Asymptotics obtained from Cauchy integral formula.
Differential geometry component:
• Analyze Jacobi equation (using Teich input) to bound first and secondderivatives of geodesic flow in terms of distance to ∂T .
Smooth ergodic theory:
• Modified Hopf argument: key property is absolute continuity. Onenovelty: Stable manifolds are complete! (thanks to geodesic convexity).
Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 32 / 32