Physical Measures. Stefano Luzzatto Abdus Salam International Centre for Theoretical Physics Trieste, Italy.

Transcription

1 Physical Measures Stefano Luzzatto Abdus Salam International Centre for Theoretical Physics Trieste, Italy. International conference on Dynamical Systems Hammamet, Tunisia September 5-7, 2017

2 Let f : M M. For x M let O + (x) := {f n (x)} n N be the orbit of x. Define a sequence of probability measures µ n (x) := 1 n Question: Does µ n (x) converge? Example (Convergence) 1 f n (x) p implies µ n (x) δ p n 1 δ f i (x) 2 µ ergodic invariant implies µ n (x) µ for µ a.e. x (Birkhoff). i=0 Example (Non-Convergence - Bowen s eye) subsequences n i, n j such that µ ni (x) δ p1 and µ nj (x) δ p2 Stefano Luzzatto (ICTP) 2 / 14

3 Let µ be a probability measure. The basin of µ is B µ := {x X : µ n (x) µ}. Definition µ is a physical measure if Leb(B µ ) > 0. Example (Physical measures) 1 δ p where p is an attracting fixed point; 2 µ invariant, ergodic, and µ Leb (µ(b µ ) = 1 Leb(B µ ) > 0); 3 Sinai-Ruelle-Bowen (SRB) measures. Example (No physical measures) 1 The identity map; 2 Bowen s eye. Conjecture (Palis conjecture) Typical systems have (finitely many) physical measures. Stefano Luzzatto (ICTP) 3 / 14

4 Conjecture (Palis conjecture) Typical systems have (finitely many) physical measures. Definition (Hyperbolicity) A measurable invariant set Λ M is (nonuniformly) hyperbolic if: 1 there exists an invariant measurable splitting T x M = Ex s Ex u s.t. ) lim (E s n ± f n (x), Eu f n (x) = 0 2 there exists χ > 0 such that for every v s E s x, v u E u x we have 1 lim n ± n ln Df x n (v s 1 ) < χ and lim n ± n ln Df x n (v u ) > χ. Λ is uniformly hyperbolic if the splitting is continuous, (Ex, s Ex) u is uniformly bounded, and 1/n ln Dfx n (v s/u ) converge uniformly in x. Conjecture (Viana conjecture) Hyperbolic systems have (finitely many) physical measures. Stefano Luzzatto (ICTP) 4 / 14

5 Existence of SRB physical measures in higher dimensions: 1 E u uniform, E s uniform, splitting continous. Sinai-Ruelle-Bowen, 1960 s-1970 s 2 E u uniform, E s non-uniform, splitting continous. Pesin-Sinai, Erg. Th. & Dyn. Syst Bonatti-Viana, Israel J. Math E u non-uniform, E s uniform, splitting continous. Alves-Bonatti-Viana, Inv. Math Alves-Dias-L-Pinheiro, J. Eur. Math. Soc E u non-uniform, E s trivial (non-uniformly expanding) Alves-L.-Pinheiro, Ann. IHP, 2005 Alves-Dias-L. Ann. IHP, E s, E u non-uniform, splitting measurable. Climenhaga-Dolgopyat-Pesin, Comm. Math. Phys Climenhaga-L.-Pesin, In progress, 2017 Remark: most results assume implicitly or explicitly additional technical conditions including recurrence assumptions. Stefano Luzzatto (ICTP) 5 / 14

6 Three methods for construction of SRB physical measures: 1 Symbolic coding and thermodynamical formalism; 2 Pushing forward Lebesgue measure; 3 Inducing (+ combination of previous two). Inducing: Let Γ M, τ : Γ N a return time s.t. f τ(x) (x) Γ. Then ˆf := f τ : Γ Γ is the corresponding induced map. The induced map ˆf may have better properties than the original map f and may be easier to study. In particular we may be able to show that ˆf admits an SRB measure. Proposition Suppose ˆf : Γ Γ admits an SRB probability ˆµ and ˆτ := τdˆµ <. Then f admits an SRB probability measure. Problem: find GOOD induced map with INTEGRABLE return times Stefano Luzzatto (ICTP) 6 / 14

7 Example: Pomeau-Manneville maps: Let γ 0 and f(x) = x + x 1+γ mod 1. For γ = 0, f(x) = 2x mod 1 and Lebesgue is ergodic and invariant. For γ > 0, f (x) = 1 + (1 + γ)x γ and so f (0) = 1. Since 0 is a fixed point, f is not uniformly expanding and we cannot apply any existing methods to construct physical measures. The inducing time is integrable if and only if γ [0, 1). Stefano Luzzatto (ICTP) 7 / 14

8 Example: Anosov Diffeomorphism Γ = element of the Markov partition; τ = first return time fk fk fk fk fn fn F = f τ : Γ Γ is a Young Tower and therefore µ SRB for F. Moreover, τ integrable and therefore µ SRB for f. Similar construction without a priori Markov partitions (Young 98) Stefano Luzzatto (ICTP) 8 / 14

9 Definition (Rectangle) Let Λ be a hyperbolic set. A set Γ Λ is a rectangle if for every x, y Γ, the intersection Vx s Vy u is a single point which belongs to Γ. W u (P) P W s (Q) R W s (P) Q W u (Q) Γ = C S C U where C S = x Γ V s (x), C U = x Γ V u (x) In general Γ is a Cantor set so the Young Tower is harder to get. Stefano Luzzatto (ICTP) 9 / 14

10 W u (P) P W s (Q) R W s (P) Q W u (Q) We let Γ denote the region bounded by V s p, V u p, V s q, V u q. Definition A rectangle Γ is 1 nice if f i (V s p/q ) int Γ = and f i (V u p/q ) int Γ = i > 0. 2 recurrent if every x Γ returns to Γ; 3 fat if Leb( x Γ0 V s x ) > 0. Stefano Luzzatto (ICTP) 10 / 14

11 Definition A rectangle Γ is 1 nice if f i (V s p/q ) int Γ = and f i (V u p/q ) int Γ = i > 0. 2 recurrent if every x Γ returns to Γ; 3 fat if Leb( x Γ0 V s x ) > 0. Theorem (Climenhaga-L-Pesin) Let f : M M be a C 2 surface diffeomorphism with a 1 hyperbolic set Λ M, and a 2 nice recurrent rectangle Γ Λ. Then there exists a Topological First Return Young Tower. If Γ is fat then this Topological Young Tower satisfies hyperbolicity and distortion estimates and f admits an SRB probability measure. Stefano Luzzatto (ICTP) 11 / 14

12 Definition (Hyperbolic branch) A hyperbolic branch f i : Ĉs Ĉu is a stable strip that gets mapped to an unstable strip with uniformly hyperbolic estimates. p bc U bc S b 0 q Stefano Luzzatto (ICTP) 12 / 14

13 Definition (Hyperbolic recurrence) A rectangle Γ is hyperbolically recurrent if every return to Γ gives rise to a hyperbolic branch. Theorem Let Γ 0 Λ be a nice recurrent rectangle. Then Γ 0 is a nice hyperbolically recurrent rectangle. Definition (Saturation) A hyperbolically recurrent rectangle Γ = C s C u is saturated if for every hyperbolic branch f i : Ĉs Ĉu we have f i (Ĉu C s ) C s and f i (Ĉs C u ) C u. Theorem Let Γ 0 be a nice hyperbolically recurrent rectangle. Then there is a nice hyperbolically recurrent saturated rectangle Γ. Stefano Luzzatto (ICTP) 13 / 14

14 Theorem Let Γ be a nice hyperbolically recurrent saturated rectangle. Then Γ admits a First Return Topological Young Tower. Theorem Let Γ be a fat nice hyperbolically recurrent saturated rectangle. Then Γ admits a First Return Young Tower. Theorem Let Λ be a hyperbolic set. Let Γ Λ be a fat nice hyperbolically recurrent saturated rectangle. Then Γ admits a Young Tower with Integrable Return Times. Stefano Luzzatto (ICTP) 14 / 14