dynamics, geometry and topologyspectral geometry topology author, another short paper title dynamics...
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DYNAMICS, Geometry and Topology
Dan Burghelea (and Stefan Haller)
Department of mathematicsOhio State University, Columbuus, OH
Goettingen, November, 2010
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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
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CONTENTS
Dynamics
Spectral geometry
Topology
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Dynamics = Smooth vector field X on a closed manifold M.
Elements of Dynamics=
Rest pointsVisible trajectories
1. Instantons,
2. Closed trajectories
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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
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PROBLEM:Count rest points, instantons and closed trajectories,Relate the results of the counting to geometry andtopology,Provide new invariants for a vector field,
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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) .
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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.
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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)
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>>
>
>
Red – Stable manifold Blue – Unstable manifold
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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).
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^ ^^ ^ ^ ^
No Instantons
^ ^
^^
^^
Four Instantons
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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)
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o(s) F(s, r -- 6 (s))
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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]) .
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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
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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
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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?
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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).
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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.
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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.
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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.
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• 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.
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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
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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).
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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.
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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)).
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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
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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 .
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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.
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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.
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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
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• 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:
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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))
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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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