SRB measures for non-uniformly hyperbolic systems

Transcription

1 SRB measures for non-uniformly hyperbolic systems Vaughn Climenhaga University of Maryland October 21, 2010 Joint work with Dmitry Dolgopyat and Yakov Pesin

2 1 and classical results Definition of SRB measure Some known results 2 Decomposing the space of invariant measures Recurrence to compact sets 3 Sequences of local diffeomorphisms Frequency of large admissible manifolds 4 Usable hyperbolicity Existence of an SRB measure

3 Definition of SRB measure Some known results Physically meaningful invariant measures M a compact Riemannian manifold f : M M a C 1+ε local diffeomorphism M the space of Borel measures on M M(f ) = {µ M µ is f -invariant}

4 Definition of SRB measure Some known results Physically meaningful invariant measures M a compact Riemannian manifold f : M M a C 1+ε local diffeomorphism M the space of Borel measures on M M(f ) = {µ M µ is f -invariant} Birkhoff ergodic theorem. If µ M(f ) is ergodic then it describes the statistics of µ-a.e. trajectory of f : for every integrable ϕ, 1 n 1 lim ϕ(f k (x)) = n n k=0 ϕ dµ

5 Definition of SRB measure Some known results Physically meaningful invariant measures M a compact Riemannian manifold f : M M a C 1+ε local diffeomorphism M the space of Borel measures on M M(f ) = {µ M µ is f -invariant} Birkhoff ergodic theorem. If µ M(f ) is ergodic then it describes the statistics of µ-a.e. trajectory of f : for every integrable ϕ, 1 n 1 lim ϕ(f k (x)) = n n k=0 ϕ dµ To be physically meaningful, a measure should describe the statistics of Lebesgue-a.e. trajectory.

6 SRB measures Definition of SRB measure Some known results Smooth/absolutely continuous invariant measures are physically meaningful, but...

7 SRB measures Definition of SRB measure Some known results Smooth/absolutely continuous invariant measures are physically meaningful, but......many systems are not conservative. Interesting dynamics often happen on a set of Lebesgue measure zero.

8 SRB measures Definition of SRB measure Some known results Smooth/absolutely continuous invariant measures are physically meaningful, but......many systems are not conservative. Interesting dynamics often happen on a set of Lebesgue measure zero. absolutely continuous a.c. on unstable manifolds µ M(f ) is an SRB measure if 1 all Lyapunov exponents non-zero; 2 µ has a.c. conditional measures on unstable manifolds. SRB measures are physically meaningful. Goal: Prove existence of an SRB measure.

9 Uniform geometric structure Definition of SRB measure Some known results SRB measures are known to exist in the following settings. Uniformly hyperbolic f (Sinai, Ruelle, Bowen) Partially hyperbolic f with positive/negative central exponents (Alves Bonatti Viana, Burns Dolgopyat Pesin Pollicott) Key tool is a dominated splitting T x M = E s (x) E u (x). 1 E s, E u depend continuously on x. 2 (E s, E u ) is bounded away from 0. Both conditions fail for non-uniformly hyperbolic f.

10 Non-uniformly hyperbolic maps Definition of SRB measure Some known results The Hénon maps f a,b (x, y) = (y + 1 ax 2, bx) are a perturbation of the family of logistic maps g a (x) = 1 ax 2. 1 g a has an absolutely continuous invariant measure for many values of a. (Jakobson) 2 For b small, f a,b has an SRB measure for many values of a. (Benedicks Carleson, Benedicks Young) 3 Similar results for rank one attractors small perturbations of one-dimensional maps with non-recurrent critical points. (Wang Young) Genuine non-uniform hyperbolicity, but only one unstable direction, and stable direction must be strongly contracting.

11 Constructing invariant measures Decomposing the space of invariant measures Recurrence to compact sets f acts on M by f : m m f 1. Fixed points of f are invariant measures. Césaro averages + weak* compactness invariant measures: µ n = 1 n 1 n k=0 f k m µ nj µ M(f )

12 Constructing invariant measures Decomposing the space of invariant measures Recurrence to compact sets f acts on M by f : m m f 1. Fixed points of f are invariant measures. Césaro averages + weak* compactness invariant measures: µ n = 1 n 1 n k=0 f k m µ nj µ M(f ) Idea: m = volume µ is an SRB measure. H = {x M all Lyapunov exponents non-zero at x} S = {ν M ν(h) = 1, ν a.c. on unstable manifolds} S M(f ) = {SRB measures} S is f -invariant, so µ n S for all n. S is not compact. So why should µ be in S?

13 Non-uniform hyperbolicity in M Decomposing the space of invariant measures Recurrence to compact sets Theme in NUH: choose between invariance and compactness. Replace unstable manifolds with n-admissible manifolds V: d(f k (x), f k (y)) Ce λk d(x, y) for all 0 k n and x, y V.

14 Non-uniform hyperbolicity in M Decomposing the space of invariant measures Recurrence to compact sets Theme in NUH: choose between invariance and compactness. Replace unstable manifolds with n-admissible manifolds V: d(f k (x), f k (y)) Ce λk d(x, y) for all 0 k n and x, y V. S n = {ν supp. on and a.c. on n-admissible manifolds, ν(h) = 1}. This set of measures has various non-uniformities. 1 Value of C, λ in definition of n-admissibility. 2 Size and curvature of admissible manifolds. 3 ρ and 1/ρ, where ρ is density wrt. leaf volume.

15 Non-uniform hyperbolicity in M Decomposing the space of invariant measures Recurrence to compact sets Theme in NUH: choose between invariance and compactness. Replace unstable manifolds with n-admissible manifolds V: d(f k (x), f k (y)) Ce λk d(x, y) for all 0 k n and x, y V. S n = {ν supp. on and a.c. on n-admissible manifolds, ν(h) = 1}. This set of measures has various non-uniformities. 1 Value of C, λ in definition of n-admissibility. 2 Size and curvature of admissible manifolds. 3 ρ and 1/ρ, where ρ is density wrt. leaf volume. Given K > 0, let S n (K) be the set of measures for which these non-uniformities are all controlled by K. large K worse non-uniformity S n (K) is compact, but not f -invariant.

16 Decomposing the space of invariant measures Recurrence to compact sets Conditions for existence of an SRB measure M be a compact Riemannian manifold, U M open, f : U M a local diffeomorphism with f (U) U. Let µ n be a sequence of measures whose limit measures are all invariant. Fix K > 0, write µ n = ν n + ζ n, where ν n S n (K). Theorem (C. Dolgopyat Pesin, 2010) If lim n ν n > 0, then some limit measure of {µ n } has an ergodic component that is an SRB measure for f.

17 Decomposing the space of invariant measures Recurrence to compact sets Conditions for existence of an SRB measure M be a compact Riemannian manifold, U M open, f : U M a local diffeomorphism with f (U) U. Let µ n be a sequence of measures whose limit measures are all invariant. Fix K > 0, write µ n = ν n + ζ n, where ν n S n (K). Theorem (C. Dolgopyat Pesin, 2010) If lim n ν n > 0, then some limit measure of {µ n } has an ergodic component that is an SRB measure for f. The question now becomes: How do we obtain recurrence to the set S n (K)?

18 Coordinates in TM Sequences of local diffeomorphisms Frequency of large admissible manifolds We use local coordinates to write the map f along a trajectory as a sequence of local diffeomorphisms. {f n (x) n 0} is a trajectory of f U n T f n (x)m is a neighbourhood of 0 small enough so that the exponential map exp f n (x): U n M is injective f n : U n R d = T f n+1 (x)m is the map f in local coordinates

19 Coordinates in TM Sequences of local diffeomorphisms Frequency of large admissible manifolds We use local coordinates to write the map f along a trajectory as a sequence of local diffeomorphisms. {f n (x) n 0} is a trajectory of f U n T f n (x)m is a neighbourhood of 0 small enough so that the exponential map exp f n (x): U n M is injective f n : U n R d = T f n+1 (x)m is the map f in local coordinates Suppose R d = T f n (x)m has an invariant decomposition E u n E s n with asymptotic expansion (contraction) along E u n (E s n). Df n (0) = A n B n f n = Df n (0) + s n f n (v, w) = (A n v + g n (v, w), B n w + h n (v, w))

20 Controlling hyperbolicity and regularity Sequences of local diffeomorphisms Frequency of large admissible manifolds R d = E u n E s n f n = (A n B n ) + s n Start with an admissible manifold V 0 tangent to E0 u at 0, push it forward and define an invariant sequence of admissible manifolds by V n+1 = f n (V n ). V n = graphψ n = {v + ψ n (v)} ψ n : B(E u n, r n ) E s n Need to control the size r n and the regularity Dψ n, ψ n ε.

21 Controlling hyperbolicity and regularity Sequences of local diffeomorphisms Frequency of large admissible manifolds R d = E u n E s n f n = (A n B n ) + s n Start with an admissible manifold V 0 tangent to E0 u at 0, push it forward and define an invariant sequence of admissible manifolds by V n+1 = f n (V n ). V n = graphψ n = {v + ψ n (v)} ψ n : B(E u n, r n ) E s n Need to control the size r n and the regularity Dψ n, ψ n ε. Consider the following quantities: λ u n = log( A 1 n 1 ) λ s n = log B n α n = (En u, En) s C n = s n C 1+ε

22 Classical Hadamard Perron results Sequences of local diffeomorphisms Frequency of large admissible manifolds Uniform case: Constants such that λ s n λ s < 0 < λ u < λ u n α n ᾱ > 0 C n C < Then V n has uniformly large size: r n r > 0.

23 Classical Hadamard Perron results Sequences of local diffeomorphisms Frequency of large admissible manifolds Uniform case: Constants such that λ s n λ s < 0 < λ u < λ u n α n ᾱ > 0 C n C < Then V n has uniformly large size: r n r > 0. Non-uniform case: λ s n, λ u n, α n still uniform, but C n not. C n grows slowly r n decays slowly

24 Classical Hadamard Perron results Sequences of local diffeomorphisms Frequency of large admissible manifolds Uniform case: Constants such that λ s n λ s < 0 < λ u < λ u n α n ᾱ > 0 C n C < Then V n has uniformly large size: r n r > 0. Non-uniform case: λ s n, λ u n, α n still uniform, but C n not. C n grows slowly r n decays slowly We want to consider the case where λ s n < 0 < λ u n may fail (may even have λ u n < λ s n) α n may become arbitrarily small C n may become arbitrarily large (no control on speed)

25 Usable hyperbolicity Sequences of local diffeomorphisms Frequency of large admissible manifolds In order to define ψ n+1 implicitly, we need control of the regularity of ψ n. Control Dψ n and Dψ n ε by decreasing r n if necessary. So how do we guarantee that r n becomes large again?

26 Usable hyperbolicity Sequences of local diffeomorphisms Frequency of large admissible manifolds In order to define ψ n+1 implicitly, we need control of the regularity of ψ n. Control Dψ n and Dψ n ε by decreasing r n if necessary. So how do we guarantee that r n becomes large again? β n = C n (sinα n+1 ) 1 Fix a threshold value β and define the usable hyperbolicity: { ( min λ u n, λ u n + 1 ε λ n = (λu n λ s n) ) ( ) if β n β, min λ u n, λ u n + 1 ε (λu n λ s n), 1 βn ε log β n+1 if β n > β.

27 A Hadamard Perron theorem Sequences of local diffeomorphisms Frequency of large admissible manifolds Write F n = f n 1 f 1 f 0 : U 0 R d = T f n (x)m. Let V 0 R d be a C 1+ε manifold tangent to E u 0 at 0, and let V n(r) be the connected component of F n (V 0 ) (B(E u n, r) E s n) containing 0. Theorem (C. Dolgopyat Pesin, 2010) 1 Suppose β and χ > 0 are such that lim n 1 n n k=0 λ k > χ > 0. Then there exist constants γ, κ, r > 0 and a set Γ N with positive lower asymptotic frequency such that for every n Γ, 1 V n ( r) is the graph of a C 1+ε function ψ n : B E u n ( r) E s n satisfying Dψ n γ and Dψ n ε κ; 2 if F n (x), F n (y) V n ( r), then for every 0 k n, F n (x) F n (y) e k χ F n k (x) F n k (y).

28 Cone families Usable hyperbolicity Existence of an SRB measure Return to a local diffeomorphism f : U M. Given x M, a subspace E T x M, and an angle θ, we have a cone K(x, E, θ) = {v T x M (v, E) < θ}. If E, θ depend measurably on x, this defines a measurable cone family. Suppose A U has positive Lebesgue measure, is forward invariant, and has two measurable cone families K s (x), K u (x) s.t. 1 Df (K u (x)) K u (f (x)) for all x A 2 Df 1 (K s (f (x))) K s (x) for all x f (A) 3 T x M = E s (x) E u (x)

29 Usable hyperbolicity (again) Usable hyperbolicity Existence of an SRB measure Define λ u, λ s : A R by λ u (x) = inf{log Df (v) v K u (x), v = 1}, λ s (x) = sup{log Df (v) v K s (x), v = 1}. Let α(x) be the angle between the boundaries of K s (x) and K u (x). Fix ᾱ > 0 and consider the quantities { 1 ε α(f (x)) log α(x) if α(x) < ᾱ, ζ(x) = + if α(x) ᾱ. { λ(x) = min λ u (x), λ u (x) + 1 } ε (λu (x) λ s (x)), ζ(x)

30 An existence result Usable hyperbolicity Existence of an SRB measure Consider points with positive asymptotic usable hyperbolicity: { } 1 n 1 S = x A lim λ(f k 1 n 1 (x)) > 0 and lim λ s (f k (x)) < 0. n n n n k=0 k=0

31 An existence result Usable hyperbolicity Existence of an SRB measure Consider points with positive asymptotic usable hyperbolicity: { } 1 n 1 S = x A lim λ(f k 1 n 1 (x)) > 0 and lim λ s (f k (x)) < 0. n n n n k=0 Theorem (C. Dolgopyat Pesin, 2010) k=0 If there exists ᾱ > 0 such that LebS > 0, then f has a hyperbolic SRB measure supported on Λ.

32 An existence result Usable hyperbolicity Existence of an SRB measure Consider points with positive asymptotic usable hyperbolicity: { } 1 n 1 S = x A lim λ(f k 1 n 1 (x)) > 0 and lim λ s (f k (x)) < 0. n n n n k=0 Theorem (C. Dolgopyat Pesin, 2010) k=0 If there exists ᾱ > 0 such that LebS > 0, then f has a hyperbolic SRB measure supported on Λ. Theorem (C. Dolgopyat Pesin, 2010) Fix x U. Let V(x) be an embedded submanifold such that T x V(x) K u (x), and let m V be leaf volume on V(x). Suppose that there exists ᾱ > 0 such that lim r 0 m V (S B(x, r)) > 0. Then f has a hyperbolic SRB measure supported on Λ.