DYNAMICS, Geometry and TopologySpectral geometry Topology Author, Another Short Paper Title Dynamics = Smooth vector ﬁeld X on a closed manifold M: Elements of Dynamics= Rest points

DYNAMICS, Geometry and Topology

Dan Burghelea (and Stefan Haller)

Department of mathematicsOhio State University, Columbuus, OH

Goettingen, November, 2010

Author, Another Short Paper Title

Prior work:

S. Smale, D.Fried, S.P.Novikov, A.V. Pajitnov,M. Hutchins- Y. J. Lee

Influential work :

E Witten, B.Hellfer - J.Sjöstrandt, T.Kato, M. Shubin

Work of Burghelea- Haller reported in this talk:1. Dynamics, Laplace transform and spectral geometry,Journal of Topology, No 1 20082. Torsion, as a function on the space of representations, inC*algebras and Elliptic Theory II, Trends in Mathematics, 20083. Papers in preparation


CONTENTS

Dynamics

Spectral geometry

Topology


Dynamics = Smooth vector field X on a closed manifold M.

Elements of Dynamics=

Rest pointsVisible trajectories

1. Instantons,

2. Closed trajectories


Under nondegeneracy hypothesis = Property G

1 The set of rest points is finite2 The set of instantons is at most countable,3 The set of closed trajectories is at most countable.

Theorem(Kupka -Smale) The set of vector fields which satisfy G isresidual in any Cr−topology


PROBLEM:Count rest points, instantons and closed trajectories,Relate the results of the counting to geometry andtopology,Provide new invariants for a vector field,


Crash course in Dynamics

• M a smooth closed manifold

• X : M → TM a smooth vector field

• Ψt : M → M the flow of X a one parameter group ofdiffeomorphisms.

• θm(t) := Ψt (m) the trajectory with θm(0) = m.

• A parametrized trajectory θ : R→ M, is characterized by

dθ(t)/dt = X (θ(t) .


REST POINTS:

X := x ∈ M|X (x) = 0

For x ∈ X consider Dx (X ) : Tx (M)→ Tx (M)

the composition TxM Dx X→ Tx (TM)pr→ TxM.

• x ∈ X non degenerate= hyperbolic if

λ ∈ Spec Dx (X )⇒ <λ 6= 0.

• x ∈ X non degenerate⇒ has Morse index

indx = ]λ ∈ Spec Dx (X )|<λ > 0.



STABLE/UNSTABLE SET

for x ∈ X

W±x := y ∈ M| lim

t→±∞θx (t) = x

If x ∈ Xq then W±x are images of one to one immersions

W−x =im(i−x : Rq → M)

W +x =im(i−x : Rn−q → M)


>>

>

>

Red – Stable manifold Blue – Unstable manifold


TRAJECTORIES

• parametrized trajectory: θ : R→ M

• equivalency of parametrized trajectories: θ1 ≡ θ2iff θ1(t) = θ2(t + A) (for some A ∈ R)

• nonparametrized trajectory: is an equivalency class [θ] ofparametrized trajectories.

• instanton between two rest points x and y : is an Isolatednon parametrized trajectory [θ], with

limt→−∞ θ(t) = x and limt→∞ θ(t) = y

• closed trajectory : is a pair ˆ[θ] = ([θ],T ) s.t θ(t + T ) = θ(t).


^ ^^ ^ ^ ^

No Instantons

^ ^

^^

^^

Four Instantons


Definition• An instanton [θ] (from x to y , x , y non degenerate rest points)is non degenerate if i−x t i+y along [θ], in which case

index x − index y = 1.

• Orientations ox and oy of the unstable manifolds W−x , and

W−y provide a sign

εox ,oy ([θ]) = ±1

For a closed trajectory ˆ[θ] = ([θ],T ) and m ∈ Γ, (Γ = θ(R),)consider:

Dm(ΨT ) : Tm(M)→ Tm(M) and

Dm(ΨT ) : Tm(M)/Tm(Γ)→ Tm(M)/Tm(Γ)


o(s) F(s, r -- 6 (s))


Definition

The closed trajectory ˆ[θ] = ([θ],T ) is non degenerate iff

(Id − Dm(ΨT ))

is invertible; if soε([ ˆ[θ]]) := sign det(Id − DΨT (m))

p( ˆ[θ]) = sup k so that ˆ[θ] : S1 → M factors by a self mapof S1 of degree k .Assign to ˆ[θ] the rational number

ε( ˆ[θ])/p( ˆ[θ]) .


Property G= H + MS + NCT

1 Property H: All rest points are hyperbolic.2 Property MS: For any two rest points x , y ∈ X i−x t i+y .3 Property NCT: All closed trajectories are nondegenerate.

Denote by :

C the set of closed trajectories andIx ,y the set of instantons from x to y .

For X satisfying G the sets C and Ix ,y are countable andnaturally filtered by finite sets


Choose ξ ∈ H1(M; C) and ω ∈ Ω1(M),dω = 0 representing ξ.For a closed trajectory [θ] define the functions

Z ξ

[θ](z) :=

ε([θ])

p([θ])ez

R[θ] ξ

and for an instanton from x to y the function

Iox ,oy ,ω

[θ] (z) := εox ,oy ([θ])ezR[θ] ω


and then the formal power series

Z ξ(z) =∑

[θ]∈Cx,y

Z[θ](z)

Iox ,oy ,ω

[θ] (z) =∑

[θ]∈Ix,y

Iox ,oy[θ] (z)

WHEN THE SUMS DEFINED ABOVE ARE ACTUALLYHOLOMORPHIC FUNCTIONS?


A Riemannian metrics on M induces Riemannian metrics onW−

x if x nondegenerate. Denote by Bx (R) the ball of radius Rin W−

x .

DefinitionX satisfies (EG) (exponential growth property) if w.r. to some

(and then with any) Riemannian metric if Vol(Bx (R)) ≤ eCR

(for some C > 0).


Definition

A closed 1-form ω ∈ Ω1(M) is called Lyapunov for X ifω(X )(m) ≤ 0 and ω(X )(m) = 0 only when m ∈ X .

A cohomology class [ω] is Lyapunov for X if it contains aclosed 1-form Lyapunov fo X .

X satisfies (L) (Lyapunov property)if the set of cohomologyclasses Lyapunov for X is nonempty.


TheoremSuppose X satisfies L and EG, ξ is a Lyapunov cohomologyclass for X and ω ∈ ξ. Choose a collection of orientations of theunstable manifolds O = ox , x ∈ X.

1. There exists ρ ∈ R so that the formal sums Z ξ(z) andIox ,oy ,ω

[θ] (z) are absolutely convergent for <z > ρ and defineholomorphic functions in z ∈ C|<z > ρ.

2. For any x ∈ Xk+1, z ∈ Xk−1 one has∑y∈Xk

IO,ωx ,y · IO,ωy ,z = 0.


A holomorphic family of cochain complexes

Suppose X satisfies G, L and EG. Define

C∗(M; X , ω)(z) := (C∗(M; X ), δ∗O,ω(z))

• Ck (M; X ) := Maps(Xk ,C)

• δ∗O,ω(z) : C∗(M; X )→ C∗+1(M; X ))

(δkO,ω(z)(f ))(u) :=

∑v∈X

IO,ωu,v (z)f (v)

This is a holomorphic family of cochain complexes with a base.


• If ω1 and ω2 represent ξ ∈ H1(M; R), O1 and O2 are two setsof orientations, there exists a canonical isomorphism between

(C∗(M; X ), δ∗O1,ω1(z)) and (C∗(M; X ), δ∗O2,ω2

(z)).

⇒ C∗(M; X , ξ)(z)

• The cohomology changes in dimension at finitely many z.


Riemannian geometry

M closed manifoldX smooth vector field satisfying Gω Lyapunov closed one form for X

There exists g Riemannian metric so that X = −gradgω

Introduce Witten deformation (Ω∗M,dω(z) = d + zω ∧ · · · )equipped with the scalar products on Ω(M) defined by g. Whenz is a real number It Induces Laplacians

∆ωq (z) : Ω∗(M)→ Ω∗(M) = ∆q + z(LX + L∗X ) + z2|gradgω|2


One can show that for <z > ρ

Exactly nk eigenvalues of ∆ωk (z) go to zero exponentially

fast while the remaining have real part go to∞ linearly fastas <z →∞. As a consequenceOne has a canonical orthogonal decomposition

(Ω∗(M); dω(z)) := (Ω∗(M)sm(M),dω(z))⊕(Ω∗(M)la(M),dω(z))

which diagonalizes ∆ω∗ (z) = ∆sm

∗ (z)⊕∆la∗ (z).


Theorem1. The above decomposition has an analytic continuation in theneighborhood of [0,∞).

If in addition X satisfies EG then:

2. For <z > ρ the Integration of differential forms on unstablemanifolds is well defined and provides an isomorpfism from(Ω∗(M)sm; dω(z)) to (C∗(M; X ), δ∗O1,ω1

(z)). The isomorphism iscanonical.

3. For <z > ρ large the (Ω∗(M)la; dω(z)) is acyclic and

log Tla(M,g, ω)(z)− Z X ,[ω](z) = zR(g,X , ω)

where Tla(M,g, ω)(z)is the complex Ray Singer torsion andR(g,X , ω) a numerical invariant defined below.


Other invariants

X vector field which satisfies L with nk rest points of index k ona Riemannian manifold (M,g).

Consider∆X

q (z) : Ω∗(M)→ Ω∗(M) = ∆q + z(LX + L∗X ) + z2||X ||2

Results of Kato and Shubin⇒For any x ∈ Xk one can associate a the functionλ(z) ∈ Spec´(kz), holomorphic in a neighborhood of [0,∞) withlim<z→∞ λx (z) = 0.

Consider x λx (0) and

Pk (z) =∏

x∈Xk

(z − λx (0)).


Theorem

For any k there exists a collection λkn(z) ,n ∈ N,each

holomorphic function in a neighborhood of [0,∞) so that foreach t ∈ [0,∞) the family λk

r (t), r = 1,2, · · · represent theentire family of eigenvalues of ∆X

k (t) (with their multiplicity)

Mild improvement of Witten methods shows that each rest pointx of index k provides for t very large an eigenvalue which goexponentially fast to zero. In fact much more is true


TOPOLOGY

Twisted cohomology

Is a graded vector space H∗(M; ξ) associated to(M, ξ ∈ H1(M; C).

Denote βξi := dimH i(M; ξ).

For <z large enough βzξi is constant in z.

Denote βξi := lim<z→∞ βzξi .


Theorem(Novikov)If X satisfies H, MS, L with ξ a Lyapunov class for X then:

1. For <z large enough the cohomology of the cochain complexC∗(M; X , ξ)(z) and H∗(M; zξ) are naturally isomorphic.

2. One has

nk ≥ βξk∑0≤k≤r

(−1)knk ≥∑

0≤k≤r

βξk , r even

∑0≤k≤r

(−1)knk ≥∑

0≤k≤r

βξk , r odd.


Torsion

• The cochain complex C∗(M; X , ξ)(z) is equipped with a baseso it is possible to define a torsion (in case that the zξ- twistedcohomology for is trivial is a holomorphic function in z. )

• The cohomology class zξcan be interpreted as a rank1-complex representation of the fundamental group andtogether with a so called Euler structure there is a topologicalinvariant the Reidemeister torsion (in case that the zξ- twistedcohomology for is trivial is a holomorphic function in z.)

• The vector field X and some minor additional data determinean Euler structure.


It is possible to show that the sum of the log of the torsion ofC∗(M; X , ξ)(z) and of the function Z ξ(z) is exactly theReidemeister torsion of M zξ and the Euler class defined by X .This relates the functions IO,ωx ,y (z) and Z ξ(z) and the topology


• The torsion logT ∈ C associated with a cochain complex(C∗,d∗) with a nondegerate bilinear form each C i , (hence withLaplacians ∆k : Ck → Ck ,) is defined by the formula

log T = 1/2∑i≥0

(−1)i i log det ∆i

• R(g,X , ω) ∈ R is an invariant associated with a Riemannianmanifold (M,g), a vector field X with isolated rest points and aclosed one form ω defined as follows:


For an oriented Riemannian manifold M, the Riemannianmetric produces a (n − 1) differential form (the angularmomentum form)

Ψ(g) ∈ Ωn−1(TM \M)

If the vector field X has no rest points it defines the mapX : M → T (M) \M and

R(g,X , ω) :=

∫Mω ∧ X ∗(Ψ(g))


When the vector field X has singularities the integration is doneover M \ X and the integral might not be convergent andrequires regularization. This regularization is always possible.Geometric regularization .

Since the eigenvalues of the Laplacians ∆lai are infinitely many

the definition of det ∆lai also requires regularization. This

regularization is again possible.Zeta function regularization.


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DYNAMICS, Geometry and TopologySpectral geometry Topology Author, Another Short Paper Title Dynamics = Smooth vector ﬁeld X on a closed manifold M: Elements of Dynamics= Rest points