DYNAMICS, Topology and Spectral Geometry

Transcription

1 DYNAMICS, Topology and Spectral Geometry Dan Burghelea Department of Mathematics Ohio State University, Columbuus, OH Iasi, June 21

2 Prior work on the topic: S. Smale, D.Fried, S.P.Novikov, A.V. Pajitnov, M. Hutchins- Y. J. Lee Influential work on different but related topoics: E Witten, B.Hellfer - J.Sjöstrandt, J.M.Bismut - W. Zhang Work of Burghelea- Haller reported in this talk: 1. Dynamics, Laplace transform and spectral geometry, Journal of Topology, No Torsion, as a function on the space of representations, in C*algebras and Elliptic Theory II, Trends in Mathematics, pp Birkhauser

3 CONTENTS Crash course on dynamics (The Problems) Topology (Results) Spectral geometry (Results)

4 Crash course in Dynamics M a smooth closed manifold X : M TM a smooth vector field Ψ t : M M the flow of X a one parameter group of diffeomorphisms. θ m (t) := Ψ t (m) the trajectory with θ m (0) = m. A parametrized trajectory θ : R M, is characterized by dθ(t)/dt = X(θ(t).

5 z y x

6 Elements of dynamics: Rest points Trajectories Properties of dynamics: (G) Genericty = H (non degenerate= hyperbolic rest points) + MS (Morse-Smale property) + NCT (non degenerate closed trajectories) ; almost all vector fields (L) Lyapunov ; most of the vector field which comes from Physics (EG) Exponential growth; possibly all vector fields which satisfy G and L

7 REST POINTS: X := {x M X(x) = 0} For x X consider D x (X) : T x (M) T x (M) the composition T x M Dx X T x (TM) pr T x M. x X non degenerate= hyperbolic if λ Spec D x (X) Rλ 0. If x X non degenerate define Morse index indx = {λ Spec D x (X) Rλ > 0}.

8 (x, (-x,-v) (-x,v) (-v,x)

9 STABLE/UNSTABLE SET for x X W ± x := {y M lim t ± θ x(t) = x} If x has Morse index q then W x ± are images of one to one immersions Wx =im(ix : R q M) W x + =im(ix : R n q M)

10 > > > > Red Stable manifold Blue Unstable manifold

11 TRAJECTORIES parametrized trajectory: θ : R M equivalency of parametrized trajectories: θ 1 (t) = θ 2 (t + A) (for some A R) θ 1 θ 2 iff nonparametrized trajectory: is an equivalency class [θ] of parametrized trajectories. instanton between two rest points x and y: is an Isolated non parametrized trajectory [θ], with lim θ(t) = x and lim t θ(t) = y t closed trajectory : is a pair ˆ [θ] = ([θ], T ) s.t θ(t + T ) = θ(t).

12 ^ ^ ^ ^ ^ ^ ^ No Instantons Four Instantons ^ ^ ^ ^ ^

13 Definition An instanton [θ] from x to y (x, y non degenerate rest points) is nondegenerate if W x W + y. If so index(x) index(y) = 1. Orientations o x and o y of the unstable manifolds W x, and W y provide a sign ɛ([θ]) = ±1 For a closed trajectory [θ] ˆ = ([θ], T ), the Poincare return map: m Γ = θ(r) consider D m (Ψ T ) : T m (M)/T m (Γ) T m (M)/T m (Γ)

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

15 Definition The closed trajectory ˆ [θ] = ([θ], T ) is nondegenerate if (Id D m (Ψ T )) is invertible; if so define ɛ( ˆ [θ]) := sign Det(Id DΨ T (m)) p( [θ]) ˆ := sup{k Z >0 s.t. the map [θ] ˆ : S 1 M factors through a degree k self map of S 1. } Assign to ˆ [θ] the rational number ɛ( ˆ [θ])/p( ˆ [θ]). A Riemannian metrics on M induces Riemannian metrics on W x. if x nondegenerate. Denote by B x (R) the ball of radius R in W x.

16 Properties Definition 1 X satisfies H (hyperbolicity) if all rest points are non degenerate. 2 X satisfies MS (Morse-Smale property)if all rest points are non degenerate and for any x, y X, W x W + y. 3 X satisfies NCT (nondegenerate closed trajectories) if all closed trajectories are non degenerate. 4 X satisfies (EG) (exponential growth volume) if w.r. to some (and then with any) Riemannian metric if Vol(B x (R)) e CR (for some C > 0).

17 Definition A closed 1-form ω Ω 1 (M) is called Lyapunov for X if ω(x)(m) 0 and ω(x)(m) = 0 only when m X. A cohomology class [ω] is Lyapunov for X if it contains a closed 1-form Lyapunov fo X. X satisfies (L) (Lyapunov property)if the set of cohomology classes Lyapunov for X is nonempty.

18 Theorem 1. Given a vector field X then for any r 0 there exist vector fields Y which are arbitrary closed to X in the C r topology and satisfy G (= all dynamics elements nondegenerate). 2. If X satisfies H (= nondegenerate rest points) resp. H and L(= Lyapunov) then Y can be choosen to agree with X in the neighborhood of X and satisfy G and L. 3. If X satisfies H and L then there exists vector fields Y which are arbitrary closed to X in the C 0 topology, agrees with X on the neighborhood of the rest points and satisfy G, L and EG (= Exponential growth of volumes) IMPORTANT CONJECTURE In 3. above replace C 0 with C r for any r.

19 THE ELEMENTS OF DYNAMICS ARE : REST POINTS, INSTANTONS and CLOSED TRAJECTORIES. THE PURPOSE OF THE SUBSEQUENT DISCUSSION IS TO SHOW HOW ONE CAN COUNT THEM AT LEAST IN THE CASE THE VECTOR FIELD SATISFIES G, L, EG

20 Notations For M a manifold and x, y M define : P x,y :=homotopy classes of path from x to y. [S 1, M] =homotopy classes of maps from S 1 to M Both P x,y and [S 1, M] are at most countable. For ω Ω 1 (M) closed 1 form, [ω] = ξ H 1 (M : R) define ω(α) := α ω, for α P x,y ω(γ) := [ω](γ) = ξ(γ) for γ [S 1, M]. For X a smooth vector field with non degenerate rest points choose O a collection of orientations o x of W x

21 Counting rest points, instantons, closed trajectories Suppose X satisfies G (all dynamics elements nondegenerate) The set of rest points of index q, X q is finite. Define n q := X q. For x, y X the set of instantons from x to y and the set of closed trajectories are at most countable. Suppose X satisfies G and L (Lyapunov) then: The set of instantons [θ] in α P x,y is finite. Define Ix,y X,O (α) = I x,y (α) := ɛ([θ]) [θ] α The set of closed trajectories [θ] ˆ in γ [S 1, M] is finite. Define Z X (γ) = Z(γ) := ˆ ɛ( [θ])/p( ˆ [θ]) ˆ [θ] γ

22 Laplace transform Let ω Lyapunov closed one form for X with [ω] = ξ H 1 (M; R). Laplace transforms of the counting functions Ix,y X,O and Z X are: I ω x,y(z) := Z ξ (z) := α P x,y e zω(α) I x,y (α) γ [S 1,M] e zξ(γ) Z(γ) If ω = ω + dh then I ω x,y(z) = e z(h(y) h(x)) I ω x,y(z)

23 Theorem 1. If X satisfies G and L then I ω u,v(z) absolutely convergent for Rz > ρ implies Z ξ (z) absolutely convergent for Rz > ρ. 2. If X satisfies G, L and EG then exists ρ R so that Iu,v(z) ω and Z ξ (z) are absolutely convergent, hence holomorphic functions, for Rz > ρ. Moreover for any u, w X and Rz > ρ one has Iu,v(z) ω Iv,w(z) ω = 0. v X

24 A cochain complex WHAT A TOPOLOGIST CAN DO WITH I O,X x,y (in view of 2.)? Suppose X satisfies G, L and EG. Define C (M; X, ω)(z) := (C (M; X), δ O,ω(z)) C k (M; X) := Maps(X k, C) δ O,ω (z) : C (M; X) C +1 (M; X)) (δ k O,ω(z)(f ))(u) := v X I ω u,v(z)f (v) If ω 1 and ω 2 represent ξ H 1 (M; R), O 1 and O 2 are two sets of orientations, there exists a canonical isomorphism between (C (M; X), δ O 1,ω 1 (z)) and (C (M; X), δ O 2,ω 2 (z)). C (M; X, ξ)(z)

25 Summary: DYNAMICS (M, X, ξ H 1 (M; R)) G, L COUNTING FUNCTIONS (I O,X x,y, Z X ) EG (Laplace transform w.r. to ω ξ) HOLOMORPHIC FUNCTIONS (Ix,y O,X,ω (z), Z X,ξ (z)) Cochain complex C (M; X, ξ)(z),

26 TOPOLOGY Twisted cohomology Is a graded vector space H (M; ξ) associated to (M, ξ H 1 (M; C). Denote β ξ i := dimh i (M; ξ). For Rz large enough β zξ i is constant in z. Denote β ξ := lim i Rz β zξ i.

27 Theorem (Novikov) If X satisfies H, MS, L with ξ a Lyapunov class for X then: 1. For Rz large enough the cohomology of the cochain complex C (M; X, ξ)(z) and H (M; zξ) are naturally isomorphic. 2. One has n k β ξ k ( 1) k n k β ξ k, r even 0 k r 0 k r 0 k r ( 1) k n k 0 k r β ξ k, r odd.

28 For (X 1, X 2 ) vector fields which satisfy G and have ξ as Lyapunov cohomology class, (by using the cochain complex (C (M; X, ξ)) one can define topologically the relative torsion τ(x 1, X 2, ξ) C \ 0 Theorem If X 1 and X 2 satisfy in addition EG then: 1. τ(x 1, X 2, ξ) = exp(z X 1,ξ (z) Z X 2,ξ (z)) 2. Given ξ H 1 (M; C), there exists vector fields X satisfying G, with ξ as a Lyapunov class and no closed trajectories.

29 A few comments 1. The proof of the above theorem is considerably simpler if the number of simple closed trajectories is finite in which case the hypothesis EG is superfluous. 2. If the inequalities in the Theorem (credited to Novikov) are violated when β ξ are replaced by the standard Betti numbers i β i, then the presence of closed trajectories is assured. 3. If the number of simple closed trajectories is finite then the numbers β ξ i, β i, and the number of simple closed trajectories with their index (not defined here ) can be related.

30 Differential topology/ Riemannian geometry M closed manifold X smooth vector field satisfying G ω Lyapunov closed one form for X There exists g Riemannian metric so that X = grad g ω Consider the Witten deformation (Ω M, d ω (z) = d + zω ) and Ω M equipped with the scalar products defined by g. It Induces Laplacians ω q (z) : Ω (M) Ω (M) = q + z(l X + L X ) + z2 grad g ω 2

31 Spectral geometry Theorem For Rz large there is a canonical orthogonal decomposition (Ω (M); d ω (z)) := (Ω (M) sm (M), d ω (z)) (Ω (M) la (M), d ω (z)) which diagonalizes ω (z) = sm (z) la (z). The eigenvalues of sm k (z) go to zero exponentially fast while the eigenvalues of la k (z) go to linearly fast (as Rz ). With O specified the " sm" -complex has a canonical base indexed by X s.t. dx,y(z) ω = Ix,y X,O,ω (z). The "la" - complex is acyclic with the Ray-Singer torsion satisfying: log T la (M, g, ω)(z) zr(g, X, ω) = Z X,[ω] (z).

32 The torsion logt C associated with a cochain complex (C, d ) with scalar products in each C i, (hence with Laplacians k : C k C k,) 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 Riemannian manifold (M, g), a vector field X with isolated rest points and a closed one form ω defined as follows:

33 For an oriented Riemannian manifold M, the Riemannian metric produces a (n 1) differential form (the angular momentum form) Ψ(g) Ω n 1 (TM \ M) If the vector field X has no rest points it defines the map X : M T (M) \ M and R(g, X, ω) := ω X (Ψ(g)) M

34 When the vector field X has singularities the integration is done over M \ X and the integral might not be convergent and requires regularization. This regularization is always possible. Geometric regularization. Since the eigenvalues of the Laplacians la i are infinitely many the definition of det la i also requires regularization. This regularization is again possible. Zeta function regularization.