Physical measures? To prove that the SRB measures are physical measures, some contraction in the E cs direction is needed. Giving a point x ∈ A , consider its largest Lyapunov exponent in the E cs direction: 1 λ c n log � Df n | E cs + ( x ) = lim sup x � . n →∞ Theorem (Bonatti and Viana 2000) Let A ⊂ M be an attractor on which f is partially hyperbolic with splitting T A M = E cs ⊕ E u . Assume that for any unstable manifold γ u we have λ c + ( x ) < 0 for a positive m γ u measure set of points x ∈ γ u . Then there are ergodic SRB measures µ 1 , ..., µ ℓ supported on A such that for m almost every x with ω ( x ) ⊂ A we have x ∈ B ( µ j ) for some 1 ≤ j ≤ ℓ . Moreover, if the leaves of the unstable foliation are dense in A , then there is a unique SRB measure supported on A . 18
Case E s ⊕ E cu Consider a partially hyperbolic attractor T A M = E s ⊕ E cu with a trapping region U . A priori, local unstable manifolds are not defined in this case. Consider an extension (not necessarily invariant) of the fiber bundles to a trapping region U . H ⊂ U is called nonuniformly expanding along E cu if there is c > 0 and a Riemannian metric on M such that for all x ∈ H n � 1 log � Df − 1 | E cu lim inf f j ( x ) � < − c . (NUE) n n → + ∞ j =1 H ⊂ U is called strongly nonuniformly expanding along the E cu direction if there is c > 0 and a Riemannian metric on M such that for all x ∈ H n � 1 log � Df − 1 | E cu lim sup f j ( x ) � < − c . (SNUE) n n → + ∞ j =1 19
Theorem (Alves, Bonatti, and Viana 2000) Let A = � n ≥ 0 f n ( U ) be a partially hyperbolic set with T A M = E s ⊕ E cu . Assume that there is H ⊂ U with m ( H ) > 0 on which f is SNUE along E cu . Then there are ergodic SRB measures µ 1 , ..., µ ℓ on A such that for Lebesgue almost every x ∈ H we have x ∈ B ( µ j ) for some 1 ≤ j ≤ ℓ . Moreover, if f is transitive in A , then there is a unique SRB measure on A . There are a disk D ⊂ A and sets H 1 , H 2 , H 3 · · · ⊂ D such that the weak* accumulation points of n − 1 � µ n = 1 f j ∗ ( m D | H j ) n j =0 have the “SRB property”. Moreover, there is some α > 0 such that n − 1 � µ n ( M ) = 1 m D ( H j ) ≥ α, for large n . n j =0 Difficulty: weak* accumulation points of ( µ n ) are not necessarily invariant. 20
Theorem (Alves, Dias, Luzzatto, and Pinheiro 2017) Let A = � n ≥ 0 f n ( U ) be a partially hyperbolic set with T A M = E s ⊕ E cu . Assume that there is H ⊂ U with m ( H ) > 0 on which f is NUE along E cu . Then there are ergodic SRB measures µ 1 , ..., µ ℓ on A such that for Lebesgue almost every x ∈ H we have x ∈ B ( µ j ) for some 1 ≤ j ≤ ℓ . Moreover, if f is transitive in A , then there is a unique SRB measure on A . Hence, using Birkhoff Ergodic Theorem we easily see that the following conditions are equivalent in this context: 1 there is H ⊂ U with m ( H ) > 0 on which f is NUE along E cu ; 2 there is H ⊂ U with m ( H ) > 0 on which f is SNUE along E cu ; 3 there is H ⊂ U with m ( H ) > 0 such that for x ∈ H n � 1 log � Df − 1 | E cu lim f j ( x ) � < 0 . n n → + ∞ j =1 21
Nonuniform expansion vs. Lyapunov exponents It remains an interesting question to know if SRB measures exist under the assumption that f has in H all Lyapunov exponents positive along E cu : 1 ∀ v ∈ E cu \ { 0 } . n log � Df n ( x ) v � > 0 , lim sup ( ∗ ) n →∞ Condition ( ∗ ), unlike (S)NUE, does not depend on the choice of metric. If dim( E cu ) = 1, then ( ∗ ) is equivalent to NUE. Open problem Assume f is partially hyperbolic and has all Lyapunov exponents positive along the E cu direction on a set with positive Lebesgue measure. Is f NUE along the E cu direction on a set with positive Lebesgue measure? 22
Decay of correlations The correlation of observables ϕ, ψ : M → R is defined as � � � � � � � � � Cor µ ( ϕ, ψ ◦ f n ) = ϕ ( ψ ◦ f n ) d µ − ϕ d µ ψ d µ � . � Taking ϕ and ψ the characteristic functions of Borel sets, we obtain the usual notion of mixing when Cor µ ( ϕ, ψ ◦ f n ) → 0 . We are interested in obtaining specific rates (polynomial, exponential,...) for the convergence of this quantity to zero, when n → ∞ . For this, we need to some regularity on the observables ϕ, ψ : M → R . We frequently assume that (at least) ϕ is H¨ older continuous: there is η > 0 such that | ϕ ( x − ϕ ( y ) | sup < ∞ . dist( x , y ) η x � = y Consider H = { ϕ : M → R | ϕ is H¨ older continuous } 23
Other statistical properties Here we will be focused on Decay of Correlations. Under the same approach (inducing schemes) several other statistical properties of SRB measures can be deduced: Central Limit Theorem (Young 1998; Young 1999); Large Deviations (Melbourne and Nicol 2008; Melbourne 2009); Almost Sure Invariance Principle (Melbourne and Nicol 2005; Melbourne and Nicol 2009); Local Limit Theorem (Gou¨ ezel 2005); Berry-Esseen Theorem (Gou¨ ezel 2005). 24
Case E cs ⊕ E u Let A be partially hyperbolic attractor such that T A M = E cs ⊕ E u and the leaves of F u are dense in A (in particular, the SRB in A is unique). Theorem (Dolgopyat 2000) If dim( M ) = 3 and E cs = E c ⊕ E s with E c mostly contracting, then the SRB measure has exponential decay of correlations and exponential large deviations for H¨ older continuous observables. Theorem (Castro 2002) If f has a Markov partition and E cs is mostly contracting, then the SRB measure has exponential decay of correlations and exponential large deviations for H¨ older continuous observables. Theorem (Castro 2004) If dim( E cs ) = 1 and E cs is mostly contracting, then the SRB measure has exponential decay of correlations and exponential large deviations for H¨ older continuous observables. 25
Case E s ⊕ E cu Let f : M → M be a C 2 diffeomorphism with a transitive partially hyperbolic attractor A for which T A M = E s ⊕ E cu . In particular, there is a unique ergodic SRB measure supported on A . If SNUE holds for x , then we have defined the expansion time � � n − 1 � N ≥ 1: 1 log � Df − 1 | E cu E ( x ) = min f i ( x ) � < − c , ∀ n ≥ N n i =0 Theorem (Alves and Pinheiro 2010) If there is a local unstable disk D ⊂ A such that m D {E > n } � n − α for some α > 0, then Cor µ ( ϕ, ψ ◦ f n ) � n − α +1 for all ϕ, ψ ∈ H . Theorem (Alves and Li 2015) If there is a local unstable disk D ⊂ A such that m {E > n } � e − cn θ for some c > 0 and 0 < θ ≤ 1, then there is c ′ > 0 such that Cor µ ( ϕ, ψ ◦ f n ) � e − c ′ n θ for all ϕ, ψ ∈ H . 26
Uniformly expanding maps We say that a differentiable map f : M → M is uniformly expanding if for some choice of a Riemannian norm in M and 0 < λ < 1 such that � Df ( x ) − 1 � ≤ λ, for all x ∈ M . Example Consider T d = R d / Z d and f : T d → T d given by the quotient of a linear map having a diagonal matrix with integer eigenvalues λ 1 , . . . , λ d ≥ 2. Theorem (Krzy˙ zewski and Szlenk 1969) Let f : M → M be a C 2 uniformly expanding map. Then f has a unique SRB measure. Moreover the support of this SRB measure is equal to M and its basin has full Lebesgue measure in M . This result was proved considering averages of push-forwards of Lebesgue measure and controlling the density of limit measures. 27
Nonuniformly expanding maps A smooth map f : M → M is nonuniformly expanding (NUE) if for some choice of a Riemannian norm in M and c > 0 we have n � 1 log � Df ( f j ( x )) − 1 � < − c , lim inf (NUE) n n → + ∞ j =1 for Lebesgue almost every x ∈ M . Strong nonuniform expansion (SNUE) is defined similarly, replacing lim inf above by lim sup. Theorem (Pinheiro 2006) Let f : M → M be a nonuniformly expanding C 2 map. Then f has a finite number of SRB measures whose basins cover a full Lebesgue measure set in M . The same conclusion holds for maps with critical/singular sets under an additional assumption of slow recurrence to the critical/singular set. 28
Decay of correlations If SNUE holds almost everywhere in M , then the expansion time � � n − 1 � N ≥ 1: 1 log � Df ( f i ( x )) − 1 � < − c , E ( x ) = min ∀ n ≥ N n i =0 is defined for Lebesgue almost every x ∈ M . Theorem (Alves, Luzzatto, and Pinheiro 2005) Let µ be an SRB measure for a C 1+ SNUE map f . If m {E > n } � n − α for some α > 0, then Cor µ ( ϕ, ψ ◦ f n ) � n − α +1 for all ϕ ∈ H and all ψ ∈ L ∞ ( m ). Theorem (Gou¨ ezel 2006) Let µ be an SRB measure for a C 1+ SNUE map f . If m {E > n } � e − cn θ for some c > 0 and 0 < θ ≤ 1, then there is c ′ > 0 such that Cor µ ( ϕ, ψ ◦ f n ) � e − c ′ n θ for all ϕ ∈ H and all ψ ∈ L ∞ ( m ). 29
Example: nonuniformly expanding map Let f : M → M be a local diffeomorphism for which there are δ > 0 small, σ < 1 and B ⊂ M a (bad) domain of injectivity of f such that: 1 � Df − 1 ( x ) � < 1 + δ , for every x ∈ B ; 2 | det( Df ( x ) | > 1, for every x ∈ B ; 3 � Df − 1 ( x ) � < σ , for every x ∈ M \ B . Lemma (Alves, Bonatti, and Viana 2000) There is θ > 0 such that for Lebesgue almost every point x ∈ M � � 1 0 ≤ j < n : f j ( x ) / lim inf n # ∈ B ≥ θ. n →∞ It follows that for small δ > 0 there is c > 0 such that for Lebesgue almost every x ∈ M n − 1 � 1 log � Df ( f j ( x )) − 1 � ≤ − c . lim sup n n → + ∞ j =0 Moreover, there is a > 0 such that m {E > n } � e − an . 30
Expanding structures 31
Gibbs-Markov maps Let (∆ 0 , A , m ) be a finite measure space. We say that F : ∆ 0 → ∆ 0 is Gibbs-Markov if there exists an m mod 0 countable partition P into measurable subsets of ∆ 0 such that: 1 Markov: F maps each ω ∈ P bijectively to ∆ 0 . 2 Nonsingular: ∃ J F > 0 such that for each A ⊂ ω ∈ P � m ( F ( A )) = J F dm . A 3 Separation: for all x , y ∈ ∆ 0 there is � � n ≥ 0 : F n ( x ), F n ( y ) lie in distinct elements of P s ( x , y ) = min . 4 Bounded distortion: ∃ K > 0 and 0 < β < 1 s.t. for all x , y ∈ ω ∈ P log J F ( x ) J F ( y ) ≤ K β s ( F ( x ) , F ( y )) . 32
Exercise Let ∆ 0 be a disk in a manifold, m Lebesgue measure on ∆ 0 , P an m mod 0 countable partition of ∆ 0 and F : ∆ 0 → ∆ 0 a diffeomorphism from each ω ∈ P onto ∆ 0 . Assume that there is κ < 1 such that for m almost all x ∈ ω ∈ P � DF ( x ) − 1 � ≤ κ. and there are K , γ > 0 such that for all x , y ∈ ω ∈ P � � � � det DF ( x ) � � � ≤ K dist( F ( x ) , F ( y )) γ . log � det DF ( y ) Show that F is a Gibbs-Markov map. 33
Reference spaces Consider the space � � | ϕ ( x ) − ϕ ( y ) | F β (∆ 0 ) = ϕ : ∆ 0 → R s.t. | ϕ | β ≡ sup < ∞ . β s ( x , y ) x � = y endowed with the norm | ϕ | β + � ϕ � ∞ , and � � F + ϕ ∈ F β (∆ 0 ) : ϕ ≥ c for some c > 0 β (∆ 0 ) = . Exercise Show that F β (∆ 0 ) is relatively compact in L 1 (∆ 0 ). An F -invariant probability measure is exact ( ⇒ mixing ⇒ ergodic) if A ∈ � n ≥ 0 F − n ( A ) and ν ( A ) > 0 = ⇒ ν ( A ) = 1 . 34
Theorem 2.1 Any Gibbs-Markov map has a unique exact absolutely continuous invariant probability measure ν . Moreover, d ν/ dm belongs in F β (∆ 0 ) and there is K > 0 such that K ≤ d ν 1 dm ≤ K . The idea is to prove that the sequence of densities of the measures n − 1 � µ n = 1 F j ∗ m . n j =0 is bounded in F β (∆ 0 ), and so it has an accumulation point in L 1 (∆ 0 ). Such accumulation point is the density of an F -invariant measure. Exercise Prove Theorem 2.1. 35
Inducing schemes Consider m a measure on M and f : M → M . Given ∆ 0 ⊂ M with m (∆ 0 ) < ∞ we say that a Gibbs-Markov F : ∆ 0 → ∆ 0 is an induced map for f if there is R : ∆ 0 → N constant on each ω ∈ P such that F | ω = f R ( ω ) | ω . Theorem 2.2 If ν is the (unique) ergodic f R -invariant probability measure ≪ m | ∆ 0 , then 1 µ = � ∞ j =0 f j ∗ ( ν |{ R > j } ) is an ergodic f -invariant measure; ⇒ � ∞ 2 µ finite ⇐ ⇒ R ∈ L 1 ( m | ∆ 0 ) ⇐ j =0 m { R > j } < ∞ ; 3 f nonsingular with respect to m = ⇒ µ ≪ m . Exercise Prove Theorem 2.2. 36
Decay of correlations Consider now the case of a smooth map f : M → M , where M is a Riemannian manifold and m is Lebesgue measure on the Borel sets, and H the space of H¨ older continuous functions from M to R . Theorem (Young 1999) Assume that f has an induced Gibbs-Markov map f R with R ∈ L 1 ( m ). Then f has some some ergodic absolutely continuous invariant probability measure µ . Moreover, if gcd { R } = 1, then for all ϕ ∈ H and ψ ∈ L ∞ ( m ) 1 if m { R > n } � n − α for some α > 0, then Cor µ ( ϕ, ψ ◦ f n ) � n − α +1 ; 2 if m { R > n } � e − cn θ for some c > 0 and 0 < θ ≤ 1, then Cor µ ( ϕ, ψ ◦ f n ) � e − c ′ n θ for some c ′ > 0. If gcd { R } = k , the same conclusion holds for f k . 37
Tower extensions Consider the partition P = { ∆ 0 , i } i of ∆ 0 and R : ∆ 0 → N associated to the induced Gibbs-Markov map f R . Define the tower over ∆ 0 as � � ∆ = ( x , ℓ ): x ∈ ∆ 0 and 0 ≤ ℓ < R ( x ) , and the tower map T : ∆ → ∆ as � ( x , ℓ + 1) , if ℓ < R ( x ) − 1; T ( x , ℓ ) = ( f R ( x ) , 0) , if ℓ = R ( x ) − 1. The map π : ∆ − → M (1) f ℓ ( x ) �− → ( x , ℓ ) satisfies f ◦ π = π ◦ T . 38
The ℓ th level of the tower is the set ∆ ℓ = { ( x , ℓ ) ∈ ∆ } . The 0 th level is naturally identified with the set ∆ 0 ⊂ M . Under this identification we have T R = f R : ∆ 0 − → ∆ 0 Gibbs-Markov. The ℓ th level of the tower is a copy of { R > ℓ } ⊂ ∆ 0 . This allows us to extend the σ -algebra A and the reference measure m to the tower ∆. We also extend the separation time to ∆ × ∆, defining s ( x , y ) for x , y ∈ ∆ in the following way: if x , y ∈ ∆ ℓ , then there exist unique x 0 , y 0 ∈ ∆ 0 such that x = T ℓ ( x 0 ) and y = T ℓ ( y 0 ). Set s ( x , y ) = s ( x 0 , y 0 ) . Define s ( x , y ) = 0 for all other points x , y ∈ ∆. We consider as before � � ϕ : ∆ → R | ∃ C > 0 : | ϕ ( x ) − ϕ ( y ) | ≤ C β s ( x , y ) , ∀ x , y ∈ ∆ F β (∆) = . 39
Define for ϕ ∈ F β | ϕ ( x ) − ϕ ( y ) | C ϕ = sup . (2) β s ( x , y ) x � = y We also consider F + β (∆) = { ϕ ∈ F β (∆) | ∃ c > 0 : ϕ ≥ c } . If ϕ ∈ F + β (∆), then 1 /ϕ is bounded. Given ϕ ∈ F + β (∆), set � � � � � � 1 C + � � C ϕ , � ϕ � ∞ , ϕ = max . (3) � � ϕ ∞ Theorem 2.3 If R ∈ L 1 ( m ), then the tower map T : ∆ → ∆ has a unique ergodic invariant probability measure ν which is equivalent to m . Moreover, d ν/ dm ∈ F + β (∆) and ( T , ν ) is exact if gcd { R } = 1. Existence and uniqueness follows from Theorem 2.2. If gcd { R } > 1, then ( T , ν ) is not even mixing. 40
Back to the original dynamics Let T : ∆ → ∆ be the tower of f R , with R ∈ L 1 ( m ) and ν the T -invariant measure given by Theorem 2.3. Define µ = π ∗ ν, where π : ∆ → M is the projection given in (1), satisfying f ◦ π = π ◦ T . Exercise 1 µ is exact and µ ≪ m if f is nonsingular with respect to m . 2 Cor ν ( ϕ ◦ π, ψ ◦ π ◦ T n ) = Cor µ ( ϕ, ψ ◦ f n ) for all ϕ, ψ and n ≥ 1. 3 given ϕ ∈ H , there is some β > 0 such that ϕ ◦ π ∈ F β (∆). The proof of Young Theorem is then a consequence of: Theorem 2.4 Assume that gcd { R } = 1 . For all ϕ ∈ F β (∆) and all ψ ∈ L ∞ ( m ) 1 if m { R > n } � n − α for some α > 0 , then Cor ν ( ϕ, ψ ◦ T n ) � n − α +1 ; 2 if m { R > n } � e − cn θ for some c > 0 and 0 < θ ≤ 1 , then Cor ν ( ϕ, ψ ◦ T n ) � e − c ′ n θ for some c ′ > 0 . 41
Given ϕ ∈ L ∞ ( m ) with ϕ � = 0, define 1 ϕ ∗ = � ( ϕ + 2 � ϕ � ∞ ) d ν ( ϕ + 2 � ϕ � ∞ ) . (4) Note that ϕ ∗ is strictly positive and its integral with respect to ν is 1. Lemma 2.5 For all ϕ ∈ F β (∆) with ϕ � = 0 we have 1 ϕ ∗ ∈ F + β (∆) and 1 / 3 ≤ ϕ ∗ ≤ 3 ; 2 Cor ν ( ϕ, ψ ◦ T n ) ≤ 3 � ϕ � ∞ � ψ � ∞ | T n ∗ λ − ν | for all ψ ∈ L ∞ ( m ) , where λ is the probability measure on ∆ such that d λ/ d ν = ϕ ∗ . We have � ϕ � ∞ ≤ ϕ + 2 � ϕ � ∞ ≤ 3 � ϕ � ∞ . (5) Since ν is a probability measure, we get 1 1 1 � ≤ ( ϕ + 2 � ϕ � ∞ ) d ν ≤ . (6) 3 � ϕ � ∞ � ϕ � ∞ 42
For all x , y ∈ ∆ we have ϕ ∗ ( x ) − ϕ ∗ ( y ) ( ϕ + 2 � ϕ � ∞ ) d ν · ϕ ( x ) − ϕ ( y ) 1 � = . (7) β s ( x , y ) β s ( x , y ) Since ϕ ∈ F β (∆), then ϕ ∗ ∈ F β (∆). From (5) and (6) we get � 1 / 3 ≤ ϕ ∗ ≤ 3, and so ϕ ∗ ∈ F + β (∆). Set a = ( ϕ + 2 � ϕ � ∞ ) d ν. We have � � � � � � � Cor µ ( ϕ, ψ ◦ T n ) = � ϕ ( ψ ◦ T n ) d ν − � ϕ d ν ψ d ν � � � � � � � � � � ϕ ∗ ( ψ ◦ T n ) d ν − ϕ ∗ d ν � = a ψ d ν � � � � � � � � � � ( ψ ◦ T n ) d λ − = a ψ d ν � � � � � � � � � � ψ dT n = a ∗ λ − ψ d ν � � ≤ a � ψ � ∞ | T n ∗ λ − ν | . Observing that a ≤ 3 � ϕ � ∞ , we get Lemma 2.5. The proof of Theorem 2.4 is reduced to estimate | T n ∗ λ − ν | . 43
Convergence to equilibrium Theorem 2.6 (Young 1999; Gou¨ ezel 2006) Assume that gcd { R } = 1 . Given any measure λ such that ϕ = d λ/ dm belongs in F + β (∆) we have: 1 if m { R > n } ≤ Cn − ζ for some C > 0 and ζ > 1 , then ∗ λ − ν | ≤ C ′ n − ζ +1 for some C ′ > 0 ; | T n 2 if m { R > n } ≤ Ce − cn η for some C , c > 0 and 0 < η ≤ 1 , then ∗ λ − ν | ≤ C ′ e − c ′ n η for some C ′ , c ′ > 0 ; | T n Moreover, c ′ does not depend on ϕ and C ′ depends only on C + ϕ . Let λ and λ ′ be probability measures in ∆ whose densities with respect to m belong in F + β (∆). Define ϕ ′ = d λ ′ ϕ = d λ and dm . dm Consider the product map T × T : ∆ × ∆ → ∆ × ∆, and P = λ × λ ′ the product measure on ∆ × ∆. Let π , π ′ : ∆ × ∆ → ∆ be the projections on the first and second coordinates respectively. 44
Consider the partition Q = { ∆ ℓ, i } of ∆, and the partition Q × Q of ∆ × ∆. Note that each element of Q × Q is sent bijectively by T × T onto a union of elements of Q × Q . For each n ≥ 1, let n − 1 � ( T × T ) − i ( Q × Q ) , ( Q × Q ) n := i =0 and ( Q × Q ) n ( x , x ′ ) be the atom in ( Q × Q ) n containing ( x , x ′ ) ∈ ∆ × ∆. Since we are assuming gcd { R } = 1, then ( T , ν ) is mixing. Using that d ν/ dm is bounded, we may find n 0 ∈ N and γ 0 > 0 such that m ( T − n (∆ 0 ) ∩ ∆ 0 ) ≥ γ 0 , ∀ n ≥ n 0 . Consider � R : ∆ → Z defined as � R ( x ) = min { n ≥ 0 : T n ( x ) ∈ ∆ 0 } . 45
We introduce a sequence of times 0 ≡ τ 0 < τ 1 < τ 2 < ... in ∆ × ∆ by τ 1 ( x , x ′ ) n 0 + � R ( T n 0 ( x )) , = τ 1 + n 0 + � τ 2 ( x , x ′ ) R ( T τ 1 + n 0 ( x ′ )) , = τ 3 ( x , x ′ ) τ 2 + n 0 + � R ( T τ 2 + n 0 ( x )) , = . . . with the falls to the ground level ∆ 0 alternating between x e x ′ . We define the simultaneous return time S : ∆ × ∆ → N as � � S ( x , x ′ ) = min τ i : ( T τ i ( x ) , T τ i ( x ′ )) ∈ ∆ 0 × ∆ 0 . Since ( T , ν ) is exact, then ( T × T , ν × ν ) is ergodic, and so S is defined m × m almost everywhere. Note that S ≥ 2 n 0 and if S ( x , x ′ ) = n , then ( T × T ) n (( Q × Q ) n ( x , x ′ )) = ∆ 0 × ∆ 0 . S | ( Q×Q ) n ( x , x ′ ) = n and 46
A simplified model Assume first that JT and the densities d λ/ dm and d λ ′ / dm are constant on each element of the partition. We may write | T n ∗ λ − T n ∗ λ ′ | ∗ P − π ′ = | π ∗ ( T × T ) n ∗ ( T × T ) n ∗ P | ≤| π ∗ ( T × T ) n ∗ ( P |{ S > n } ) − π ′ ∗ ( T × T ) n ∗ ( P |{ S > n } ) | n � ∗ ( P |{ S = i } ) − π ′ | π ∗ ( T × T ) n ∗ ( T × T ) n ∗ ( P |{ S = i } ) | + i =1 � � � π ∗ ( T × T ) n ∗ ( P |{ S > n } ) − π ′ ∗ ( T × T ) n � = ∗ ( P |{ S > n } ) � n � � | T n − i ∗ ( P |{ S = i } ) − π ′ π ∗ ( T × T ) i ∗ ( T × T ) i + ∗ ( P |{ S = i } ) | ∗ i =1 In this last equality we have used that T n − i ◦ π = π ◦ ( T × T ) i T n − i ◦ π ′ = π ′ ◦ ( T × T ) i . and 47
Now, if S ( x , x ′ ) = i , then = P (( Q × Q ) i ( x , x ′ )) � � π ∗ ( T × T ) i P | ( Q × Q ) i ( x , x ′ ) ( m | ∆ 0 ) ∗ m (∆ 0 ) � � = π ′ ∗ ( T × T ) i P | ( Q × Q ) i ( x , x ′ ) . ∗ It follows that the terms in the last sum are all equal to zero, and so | T n ∗ λ − T n ∗ λ ′ | ≤ 2 P { S > n } . Taking λ ′ = ν we have T n ∗ ν = ν , and so in this simplified model we are reduced to find an upper bound for P { S > n } . 48
General case Lemma 2.7 (Young 1998) There are θ < 1 and K > 0 such that for all n ≥ 1 � � ∞ � � ∗ λ ′ � n � ≤ 2 P { S > n } + K � T n ∗ λ − T n θ i ( i + 1) P S > . i + 1 i =1 Since T ∗ ν = ν , taking λ ′ = ν we get an upper bound for | T n ∗ λ − ν | . Moreover, in the polynomial and (stretched) exponential cases, the second term in the summation above decays at the same speed of the first one. Hence, Theorem 2.6 follows from Lemma 2.8 (Young 1998; Gou¨ ezel 2006) 1 If m { R > n } � n − ζ for some ζ > 1, then P { S > n } � n − ζ +1 . 2 If m { R > n } � e − cn η for some c > 0 and 0 < η ≤ 1, then P { S > n } � e − c ′ n η for some c ′ > 0. 49
Intermittent map Consider I = [0 , 1] and f : I → I of degree 2 such that: 1 f (0) = 0, f (1) = 1; 2 f (1 / 2 − ) = 1 and f (1 / 2 + ) = 0; 3 f is C 2 and f ′ > 1 in I \ { 0 , 1 / 2 } ; 4 there is γ > 0 such that for x near 0 f ( x ) ≈ x + x 1+ γ . Theorem (Hu 2004; Liverani, Saussol, and Vaienti 1999; Young 1999) 1 f has an ergodic probability measure µ ≪ m iff γ < 1; 2 if γ < 1, then for each ϕ ∈ H and ψ ∈ L ∞ ( m ) we have Cor µ ( ϕ, ψ ◦ f n ) � 1 / n 1 /γ − 1 ; 3 if γ ≥ 1, then δ 0 is the unique physical measure of f . 50
Consider ( x n ) n the sequence in [0 , 1 / 2] defined recursively as x 1 = 1 / 2 and f ( x n +1 ) = x n , ∀ n ≥ 1 . Letting J 0 = [1 / 2 , 1] and J n = [ x n +1 , x n ] , ∀ n ≥ 1 , consider the ( m mod 0) partition of I P = { J n : n ≥ 0 } . Defininig R | J n = n + 1 for each n ≥ 0, we have f R ( J ) = I for each J ∈ P , and so an inducing scheme f R : I → I . Lemma 2.9 (Young 1999) There are 0 < β < 1 and C > 0 such that 1 ( f R ) ′ ( x ) ≥ β − 1 , for every x ∈ I \ { 0 } ; 2 log ( f n ) ′ ( x ) ( f n ) ′ ( y ) ≤ C | f n ( x ) − f n ( y ) | , for every x , y ∈ J n ; 3 m { R > n } ≈ 1 / n 1 /γ . 51
Hence, f R : I → I is a Gibbs-Markov map for any γ > 0, and so it has an invariant ergodic probability measure µ ≪ m . Also, m { R > n } ≈ 1 / n 1 /γ . For γ < 1, we have R ∈ L 1 ( m ). Young Theorem gives that f has an ergodic measure µ ≪ m with Decay of Correlations � 1 / n − 1+1 /γ . For γ ≥ 1, we still have ∞ � f j µ = ∗ ( ν |{ R > j } ) . j =0 an ergodic absolutely continuous f -invariant measure. Not finite! Proposition 2.10 If γ ≥ 1, then δ 0 is the unique physical measure of f . First of all observe that for all n ≥ 1 we have � = ν { R > n } ≈ m { R > n } ≈ n − 1 /γ . µ ( J n ) = ν J k k ≥ n It follows that d µ 1 m ( J n ) n − 1 /γ ≈ n . dm | J n ≈ (8) 52
We are going to show that for m almost every x ∈ I we have n − 1 � 1 w ∗ δ f j ( x ) − → δ 0 , n j =0 It is enough to show that for large N and small ε > 0 we have � � 1 0 ≤ j < n : f j ( x ) ∈ [0 , x N ) lim n # > 1 − ε. n →∞ Using (8), we may find N 1 ≫ N such that µ ([ x N , 1]) µ ([ x N 1 , 1]) < ε. Consider F the first return map to [ x N 1 , 1]. Exercise The measure 1 µ ([ x N 1 , 1]) µ | [ x N 1 , 1] η = is an F -invariant ergodic probability measure equivalent to m | [ x N 1 , 1]. 53
Hence, Birkhoff Ergodic Theorem gives that for m almost all x ∈ [ x N 1 , 1] � � 1 0 ≤ j < n : F j ( x ) ∈ [0 , x N ) lim n # = η ([0 , x N )) . n →∞ We have η ([0 , x N )) = 1 − η ([ x N , 1]) = 1 − µ ([ x N , 1]) µ ([ x N 1 , 1]) > 1 − ε. Observing that the fraction of time spent in (0 , x N ) under iterations by f is larger the fraction of time spent under iterations by F , we are done. 54
Hyperbolic product structures 55
Young structures Let Λ ⊂ M be a compact set. We say that Λ has a product structure if there exist a family Γ s = { γ s } of stable disks and a family Γ u = { γ u } of unstable disks such that • Λ = ( ∪ γ u ) ∩ ( ∪ γ s ); • dim γ u + dim γ s = dim M ; • each γ s and γ u meet in exactly one point; Given x ∈ Λ, let γ ∗ ( x ) denote the element of Γ ∗ containing x , for ∗ = s , u . 56
Given disks γ, γ ′ ∈ Γ u , define Θ γ,γ ′ : γ ∩ Λ → γ ′ ∩ Λ by Θ γ,γ ′ ( x ) = γ s ( x ) ∩ γ, (9) and Θ γ : Λ → γ ∩ Λ by Θ γ ( x ) = Θ γ u ( x ) ,γ ( x ) . (10) We say that the hyperbolic product structure is measurable if the maps Θ γ,γ ′ and Θ γ are measurable, for all γ, γ ∈ Γ u . 0 ∩ Γ u for some Γ s Λ 0 ⊂ Λ is called an s -subset if Λ 0 = Γ s 0 ⊂ Γ s . 0 ∩ Γ s for some Γ u Λ 0 ⊂ Λ is called a u -subset if Λ 0 = Γ u 0 ⊂ Γ u . 57
A set Λ with a measurable product structure for which (Y 1 )-(Y 5 ) below hold will be called a Young structure. (Y 1 ) Markov: ∃ pairwise disjoint s -subsets Λ 1 , Λ 2 , · · · ⊂ Λ such that ◮ m γ (Λ ∩ γ ) > 0 and m γ (Λ \ ∪ i Λ i ) ∩ γ ) = 0 for all γ ∈ Γ u ; ◮ ∀ i ∈ N ∃ R i ∈ N such that f R i (Λ i ) is a u -subset and for all x ∈ Λ i f R i ( γ s ( x )) ⊂ γ s ( f R i ( x )) f R i ( γ u ( x )) ⊃ γ u ( f R i ( x )) . and We define the recurrence time R : Λ → N and the return map f R : Λ → Λ f R | Λ i = f R i . R | Λ i = R i and The separation time for s ( x , y ) for x , y ∈ Λ is the smallest n ≥ 0 such that ( f R ) n ( x ) and ( f R ) n ( y ) lie in distinct Λ i ’s. 58
Let C > 0 and 0 < β < 1 be constants depending only on f and Λ . (Y 2 ) Contraction on stable disks : for all γ ∈ Γ s and x , y ∈ γ ◮ dist( f R ( y ) , f R ( x )) ≤ β dist( x , y ); ◮ dist( f j ( y ) , f j ( x )) ≤ C dist( x , y ), for all 1 ≤ j ≤ R ( x ). (Y 3 ) Expansion on unstable disks : for all γ ∈ Γ u , all Λ i and x , y ∈ γ ∩ Λ i ◮ dist( x , y ) ≤ β dist( f R ( y ) , f R ( x )); ◮ dist( f j ( y ) , f j ( x )) ≤ C dist( f R ( x ) , f R ( y )), for all 1 ≤ j ≤ R ( x ). (Y 4 ) Absolute continuity of Γ s : for all γ, γ ′ ∈ Γ u , the map Θ γ,γ ′ is absolutely continuous; moreover, letting ξ γ,γ ′ denote the density of (Θ γ,γ ′ ) ∗ m γ with respect to m γ ′ , we have for all x , y ∈ γ ′ ∩ Λ 1 log ξ γ,γ ′ ( x ) ξ γ,γ ′ ( y ) ≤ C β s ( x , y ) . C ≤ ξ γ,γ ′ ( x ) ≤ C and (Y 5 ) Bounded distortion : ∃ γ 0 ∈ Γ u such that for all Λ i and x , y ∈ γ 0 ∩ Λ i log det Df R | T x γ 0 ≤ C β s ( f R ( x ) , f R ( y )) . det Df R | T y γ 0 59
We say that the Young structure has integrable return times if for some (and hence for all) γ ∈ Γ u , we have � Rdm γ < ∞ . γ ∩ Λ Let H be the space of H¨ older continuous functions from M to R . Theorem 3.1 (Young 1998; Alves and Pinheiro 2008) If f has a Young structure Λ with integrable recurrence time R and gcd( R ) = 1 , then f has a unique ergodic SRB measure µ with µ (Λ) > 0 . Moreover, 1 if m γ { R > n } ≤ Cn − a for some γ ∈ Γ u and C > 0 , a > 1 , then for all ϕ, ψ ∈ H there exists C ′ > 0 such that Cor µ ( ϕ, ψ ◦ f n ) ≤ C ′ n − a +1 ; 2 if m γ { R > n } ≤ Ce − cn a for some γ ∈ Γ u and constants C , c > 0 and 0 < a ≤ 1 , then for all ϕ, ψ ∈ H there exists C ′ > 0 such that Cor µ ( ϕ, ψ ◦ f n ) ≤ C ′ e − c ′ n a . Contribution of Gou¨ ezel, Korepanov, Kosloff, and Melbourne 2019 for the simplified set of assumptions. 60
H¨ older against bounded In the non-invertible case we have Cor( ϕ, ψ ◦ f n ) ≤ C ϕ,ψ a n with a n → 0 as n → ∞ , for ϕ ∈ H and ψ ∈ L ∞ ( m ). We have for some K > 0 C ϕ,ψ = K � ϕ � η � ψ � ∞ . Lemma 3.2 If f : M → M is a diffeomorphism, then Cor µ ( ϕ, ψ ◦ f n ) → 0 does not hold for all ϕ ∈ H and all ψ ∈ L ∞ . � Take ϕ ∈ H with ϕ d µ = 0 and � ϕ � 1 � = 0. We may write for n ≥ 1 � ϕ ◦ f − n � 1 � ϕ � 1 = � ( ϕ ◦ f − n ) ψ d µ = sup � ψ � ∞ =1 � ϕ ( ψ ◦ f n ) d µ = sup � ψ � ∞ =1 Cor µ ( ϕ, ψ ◦ f n ) = sup � ψ � ∞ =1 This gives a contradiction. 61
SRB measures Theorem 3.3 The return map f R of a Young structure has a unique ergodic SRB measure ν . Moreover, the densities of its conditionals with respect to Lebesgue on unstable disks are bounded above and below by constants. Proof similar to Theorem 2.1, controlling the densities of the measures n − 1 � ν n = 1 ( f R ) j some γ ∈ Γ u . ∗ m γ u , n j =0 Theorem 3.4 If f has a Young structure Λ with integrable recurrence times, then f has a unique ergodic SRB measure with µ (Λ) > 0 . ∞ � 1 f j � ∞ µ = ∗ ( ν |{ R > j } ) . (11) j =0 ν { R > j } j =0 62
Tower extension Let f : M → M have a Young structure Λ with recurrence time R : Λ → N . As before, we define a tower � � ˆ ( x , ℓ ): x ∈ Λ and 0 ≤ ℓ < R ( x ) ∆ = , and a tower map ˆ T : ˆ ∆ → ˆ ∆ as � ( x , ℓ + 1) , if ℓ + 1 < R ( x ); ˆ T ( x , ℓ ) = ( f R ( x ) , 0) , if ℓ + 1 = R ( x ). The ℓ -level of the tower is ∆ ℓ = { ( x , ℓ ) ∈ ˆ ˆ ∆ } . The 0-level of the tower ˆ ∆ 0 is naturally identified with Λ. We have a partition of ˆ ∆ 0 into subsets ˆ ∆ 0 , i = Λ i . This gives a partition { ˆ ∆ ℓ, i } i on each level ℓ . Collecting all these sets we obtain a partition ˆ Q = { ˆ ∆ ℓ, i } ℓ, i of ˆ ∆. 63
Setting ˆ − → π : ∆ M f ℓ ( x ) ( x , ℓ ) �− → we have f ◦ π = π ◦ ˆ T . Theorem 3.5 Let f R be the return map and ˆ T the tower map of a Young structure Λ with integrable recurrence time R. If ν is the SRB measure of f R , then ∞ � 1 ˆ T j � ∞ ∗ ( ν |{ R > j } ) ˆ ν = j =0 ν { R > j } j =0 is the unique ergodic SRB measure of ˆ T. Moreover, µ = π ∗ ˆ ν is the unique ergodic SRB measure of f with µ (Λ) > 0 . π ∗ ˆ ν gives precisely the formula in (11). 64
Quotient return map Given γ 0 ∈ Γ u as in (Y 5 ), we define the quotient map of f R on γ 0 ∩ Λ F : γ 0 ∩ Λ − → γ 0 ∩ Λ Θ γ,γ 0 ◦ f R ( x ) , x �− → where γ = γ u ( f R ( x )). Proposition 3.6 F is Gibbs-Markov with respect to the m γ 0 mod 0 partition P = { γ 0 ∩ Λ 1 , γ 0 ∩ Λ 2 , . . . } of γ 0 ∩ Λ . Lemma 3.7 Let F : γ 0 ∩ Λ → γ 0 ∩ Λ be the quotient map of f R : Λ → Λ . If ν is an SRB measure of f R , then ν 0 = (Θ γ 0 ) ∗ ν is the F-invariant probability measure such that ν 0 ≪ m γ 0 . 65
Quotient tower Fix γ 0 ∈ Γ u as in (Y 5 ), and the quotient map F : γ 0 ∩ Λ → γ 0 ∩ Λ . Consider the tower map T : ∆ → ∆ of F with recurrence time R . Notice that for all i ≥ 1 R | γ 0 ∩ Λ i = R | Λ i = R i . Since γ 0 ∩ Λ ⊂ Λ, it easily follows that for all ℓ ≥ 0 we have ∆ ℓ ⊂ ˆ T = ˆ ∆ ℓ and T | ∆ . (12) Moreover, ˆ T ◦ Θ = Θ ◦ T , where ˆ Θ : ∆ − → ∆ (13) �− → ( x , ℓ ) (Θ γ 0 ( x ) , ℓ ) . Proposition 3.8 ν is the ergodic SRB measure of ˆ If ˆ T, then Θ ∗ ˆ ν is the unique ergodic T-invariant probability measure absolutely continuous with respect to m γ 0 . 66
Decay of correlations We have π ◦ ˆ Θ ◦ ˆ T = f ◦ π and T = T ◦ Θ . (14) Let ν be the unique ergodic SRB measure of ˆ ˆ T ; µ be the unique ergodic SRB measure of f with µ (Λ) > 0; ν be the unique ergodic T -invariant measure such that ν ≪ m γ 0 . By Theorem 3.5 and Proposition 3.8, we have µ = π ∗ ˆ ν and ν = Θ ∗ ˆ ν. (15) Given ϕ, ψ ∈ H , define ˆ ψ = ψ ◦ π and ϕ = ϕ ◦ π. ˆ (16) Exercise Cor µ ( ϕ, ψ ◦ f n ) = Cor ˆ ϕ, ˆ ψ ◦ ˆ T n ) . ν ( ˆ For proving Theorem 3.1, it is enough to obtain the estimates for ϕ, ˆ ψ ◦ ˆ T n ). The idea is to reduce it to a problem on the quotient Cor ν ( ˆ tower T : ∆ → ∆, and apply Theorem 2.6. 67
Given k ≥ 1, define k − 1 � T − j ˆ ˆ ˆ Q k = Q . (17) j =0 Define the discretisation ϕ k : ˆ ϕ , setting for each Q ∈ ˆ ∆ → R of ˆ Q 2 k ϕ ◦ ˆ T k ( x ): x ∈ Q } . ϕ k | Q = inf { ˆ (18) ϕ k may as well be thought of as function on ∆. Proposition 3.9 For all ϕ, ψ ∈ H and 1 ≤ k ≤ n, ϕ, ˆ ψ ◦ ˆ T n ) ≤ Cor ν ( ϕ k , ψ k ◦ T n ) Cor ˆ ν ( ˆ T k − ψ k � 1 + 2 � ψ � 0 � ˆ T k − ϕ k � 1 . + 2 � ϕ � 0 � ˆ ψ ◦ ˆ ϕ ◦ ˆ ν on ˆ � � 1 is the L 1 -norm with respect to the probability measure ˆ ∆. ν = ν , it coincides with the L 1 -norm with respect to the Since Θ ∗ ˆ probability measure ν on ∆ for functions which are constant on stable disks. 68
Step 1 ν is ˆ Since the probability measure ˆ T -invariant, we may write � � � � � � � ϕ, ˆ ψ ◦ ˆ ( ˆ ψ ◦ ˆ ˆ T n ) = � T n ) ˆ � Cor ˆ ν ( ˆ ϕ d ˆ ν − ψ d ˆ ν ϕ d ˆ ˆ ν � � � � � � � � � T n − k ˆ � ψ k ◦ ˆ � ≤ ϕ d ˆ ν − ψ k d ˆ ν ϕ d ˆ ˆ ν � � � � � � � T k − ψ k ) ◦ ˆ T n − k ˆ � ( ˆ ψ ◦ ˆ � + ϕ d ˆ ν � � � � � � � � � ( ψ k − ˆ ψ ◦ ˆ � T k ) d ˆ + ν ϕ d ˆ ˆ ν � � T k − ψ k � 1 � ϕ � 0 . ϕ, ψ k ◦ ˆ T n − k ) + 2 � ˆ ψ ◦ ˆ ≤ Cor ˆ ν ( ˆ (19) 69
Step 2 ν is ˆ T -invariant, then ˆ Note that as ˆ T ∗ preserves absolute continuity with ν . Defining ζ k as the density of the signed measure ˆ T k respect to ˆ ∗ ( ϕ k ˆ ν ) with respect to ˆ ν , � � � � � � � T n − k ˆ ϕ, ψ k ◦ ˆ � ψ k ◦ ˆ � T n − k ) = Cor ˆ ν ( ˆ ϕ d ˆ ν − ψ k d ˆ ν ϕ d ˆ ˆ ν � � � � � � � � � � ψ k ◦ ˆ � T n − k ζ k d ˆ ≤ ν − ψ k d ˆ ν ζ k d ˆ ν � � � � � � � ( ψ k ◦ ˆ � T n − k )( ˆ � + ϕ − ζ k ) d ˆ ν � � � � � � � � � � + ψ k d ˆ ν ( ˆ ϕ − ζ k ) d ˆ ν � � � � � � � ν ( ζ k , ψ k ◦ ˆ T n − k ) + 2 � ψ � 0 � � ≤ Cor ˆ ϕ − ζ k ) d ˆ ( ˆ ν � . � (20) 70
Observing that d ˆ ϕ ◦ ˆ d ˆ T k T k )ˆ T k ∗ (( ˆ ν ) ∗ ( ϕ k ˆ ν ) = ˆ ϕ and = ζ k , d ˆ ν d ˆ ν we obtain � � � � � � � � � � � d ˆ ϕ ◦ ˆ d ˆ � � � T k T k )ˆ T k � ( ˆ ϕ − ζ k ) d ˆ ν = ∗ (( ˆ ν ) − ∗ ( ϕ k ˆ ν ) � � � � � T k − ϕ k | d ˆ ϕ ◦ ˆ ≤ | ˆ ν, which together with (20) gives T k − ϕ k � 1 . ϕ, ψ k ◦ ˆ T n − k ) ≤ Cor ˆ ν ( ζ k , ψ k ◦ ˆ T n − k ) + 2 � ψ � 0 � ˆ ϕ ◦ ˆ Cor ˆ ν ( ˆ Recalling (19), it just remains to show that ν ( ζ k , ψ k ◦ ˆ T n − k ) = Cor ν ( ϕ k , ψ k ◦ T n ) . Cor ˆ (21) 71
Step 3 Observe that � � � ( ψ k ◦ ˆ T n − k ) ζ k d ˆ ψ k d ( ˆ T n − k ψ k d ( ˆ T n ν = ( ζ k ˆ ν )) = ∗ ( ϕ k ˆ ν )) . ∗ Since ψ k and ϕ k are constant on stable disks and (14), (15) hold � � � � ψ k d ( ˆ T n ψ k d (Θ ∗ ˆ T n ψ k d ( T n ( ψ k ◦ T n ) ϕ k d ν. ∗ ( ϕ k ˆ ν )) = ∗ ( ϕ k ˆ ν )) = ∗ ( ϕ k ν )) = It follows that � � ( ψ k ◦ ˆ T n − k ) ζ k d ˆ ( ψ k ◦ T n ) ϕ k d ν. ν = On the other hand, � � � � � � d ( ˆ T k ψ k d ˆ ν ζ k d ˆ ν = ψ k d ν ∗ ( ϕ k ˆ ν )) = ψ k d ν ϕ k d ν. The last two formulas yield (21). 72
Let us estimate Cor ν ( ϕ k , ψ k ◦ T n ). Consider ϕ ∗ k as in (4). Let λ ∗ k be such that d λ ∗ k / d ν = ϕ ∗ k . Lemma 2.5 gives ∗ λ ∗ Cor ν ( ϕ k , ψ k ◦ T n ) ≤ 3 � ϕ � 0 � ψ � 0 | T n k − ν | . (22) Setting λ k = T 2 k ∗ λ ∗ k and ρ = d ν/ dm γ 0 , = dT 2 k ∗ λ ∗ d λ ∗ φ k := d λ k = ϕ ∗ k k and k ρ. (23) dm γ 0 dm γ 0 dm γ 0 Take k ≈ n / 4. Since T n ∗ λ ∗ k = T n − 2 k λ k , it follows from (22) that ∗ Cor ν ( ϕ k , ψ k ◦ T n ) ≤ 3 � ϕ � 0 � ψ � 0 | T n − 2 k λ k − ν | . (24) ∗ Lemma 3.10 There exists C > 0 such that C + φ k ≤ C, for all k ≥ 1 . We apply Theorem 2.6 to get the desired conclusions on the decay of | T n − 2 k λ k − ν | , and so on the term Cor ν ( ϕ k , ψ k ◦ T n ) in Proposition 3.9, ∗ by (22). Note that n − 2 k ≈ n / 2. 73
We are left to estimate the L 1 -norms in Proposition 3.9. Define for x ∈ ˆ ∆ and k ≥ 1 b k ( x ) = # { 1 ≤ j ≤ k : ˆ T j ( x ) ∈ ˆ ∆ 0 } . Since (12) holds, we may use the same notation as for the tower T of the quotient map F . Recall that each b k is constant on stable disks. Lemma 3.11 older continuous ϕ : M → R there are C > 0 and 0 < σ < 1 For every H¨ such that for all k ≥ 1 and x ∈ ∆ we have � T k ( x )) � σ b k ( x ) + σ b k ( ˆ ϕ ◦ ˆ T k ( x ) − ϕ k ( x ) | ≤ C | ˆ . 74
Take Q ∈ ˆ Q 2 k such that x ∈ Q . It follows from (16) and (18) that there exists y ∈ Q such that for some η > 0 ϕ ◦ ˆ T k ( x ) − ϕ k ( x ) | = | ϕ ◦ π ◦ ˆ T k ( x ) − ϕ ◦ π ◦ ˆ T k ( y ) | | ˆ ≤ | ϕ | η dist( π ( ˆ T k ( x )) , π ( ˆ T k ( y ))) η . (25) Note that as x , y belong in Q ∈ ˆ Q 2 k , then b 2 k ( y ) = b 2 k ( x ). In particular, we have b k ( y ) = b k ( x ). Take z ∈ Q such that z ∈ γ u ( x ) ∩ γ s ( y ). Then, dist( π ( ˆ T k ( x )) ,π ( ˆ T k ( y ))) ≤ dist( π ( ˆ T k ( x )) , π ( ˆ T k ( z ))) + dist( π ( ˆ T k ( z )) , π ( ˆ T k ( y ))) . (26) We need to estimate the two terms on the right hand side of (26). 75
Consider ℓ ≥ 0 such that Q ⊂ ˆ ∆ ℓ . Note that the points y 0 = ˆ T − ℓ ( y ) and z 0 = ˆ T − ℓ ( z ) are in a same stable disk of ˆ ∆ 0 . Moreover, using (14) dist( π ( ˆ T k ( z )) , π ( ˆ T k ( y ))) = dist( π ( ˆ T k + ℓ ( z 0 )) , π ( ˆ T k + ℓ ( y 0 )))) = dist( f k + ℓ ( z 0 ) , f k + ℓ ( y 0 )) . (27) Set b = b k + ℓ ( y 0 ) = b k ( x ) ≥ 0 and R b = � b − 1 j =0 R (( f R ) j ( y 0 )). Note that k + ℓ = R b + j , with j < R ( f R b ( y 0 )). By (Y 2 ), dist( f k + ℓ ( z 0 ) , f k + ℓ ( y 0 )) = dist( f j ( f R b ( z 0 )) , f j ( f R b ( y 0 ))) ≤ C dist( f R b ( z 0 ) , f R b ( y 0 )) = C dist(( f R ) b ( z 0 ) , ( f R ) b ( y 0 )) . (28) Recalling that b = b k ( x ), it follows from (27), (28) and (Y 2 ) that T k ( y ))) ≤ C β b k ( x ) dist( z 0 , y 0 ) . dist( π ( ˆ T k ( z )) , π ( ˆ (29) We are left to estimate the other term on the right hand side of (26). 76
Since ˆ T k ( Q ) ∈ ˆ Q k , we have ˆ T k ( x ), ˆ T k ( z ) in the same unstable disk of some Q ′ ∈ ˆ Q k . This implies that b k ( ˆ T k ( x )) = b k ( ˆ T k ( z )) ≤ s ( ˆ T k ( x ) , ˆ T k ( z )) . Assume Q ′ ⊂ ˆ ∆ ℓ . The points x 0 = ˆ T − ℓ ( ˆ T k ( x )) and z 0 = ˆ T − ℓ ( ˆ T k ( z )) are in a same unstable disk of ˆ ∆ 0 . Set t = s ( f R ( x 0 ) , f R ( z 0 )). We have t = s ( x 0 , z 0 ) − 1 = s ( ˆ T k ( x ) , ˆ T k ( z )) − 1 ≥ b k ( ˆ T k ( x )) − 1 Using (14) and (Y 3 ), we may write dist( π ( ˆ T k ( x )) , π ( ˆ T k ( z ))) = dist( π ( ˆ T ℓ ( x 0 )) , π ( ˆ T ℓ ( z 0 ))) = dist( f ℓ ( x 0 ) , f ℓ ( z 0 )) ≤ C dist( f R ( x 0 ) , f R ( z 0 )) ≤ C β t dist(( f R ) t ( x 0 ) , ( f R ) t ( z 0 )) ≤ C β b k ( ˆ T k ( x )) − 1 dist(( f R ) t ( x 0 ) , ( f R ) t ( z 0 )) . Altogether with (25), (26) and (29) it yields Lemma 3.11 with σ = β η . 77
Define for each k ≥ 1 k − 1 � R ◦ F j . R k = (30) j =0 Proposition 3.12 Given 0 < σ < 1 , there exists C > 0 such that for all k ≥ 1 we have � � � � � R ℓ > k σ b k d ν ≤ C σ ℓ m γ 0 m γ 0 { R ≥ ℓ } + Ck . 3 ℓ ≥ 1 ℓ ≥ k / 3 It is enough to have the result with ν 0 playing the role of m γ 0 , where ν 0 is the F -invariant measure ≪ m γ 0 . We have for each n ≥ 1 � ∞ � σ b n d ν = σ k ν { b n = k } . (31) k =0 Exercise ν | ∆ 0 = ν (∆ 0 ) ν 0 (32) 78
For a point x with b n ( x ) ≥ 1, we define r n ( x ) = min { 1 ≤ j ≤ n : T j ( x ) ∈ ∆ 0 } and s n ( x ) = max { 1 ≤ j ≤ n : T j ( x ) ∈ ∆ 0 } . For each k ≥ 1 we may write � { b n = k } ⊂ { b n = k , r n = i , s n = j } 1 ≤ i ≤ j ≤ n � � � � � r n ≥ n s n ≤ 2 n { b n = k , r n = i , s n = j } ∪ ∪ = . 3 3 1 ≤ i < n / 3 2 n / 3 < j ≤ n (33) Let us now estimate the measure of the three sets in the last union. 79
Using the T -invariance of ν and (32), we have for each i ≥ 1 ν { r n = i } = ν { R ≥ i } = ν (∆ 0 ) ν 0 { R ≥ i } . Hence � � � r n ≥ n ≤ ν (∆ 0 ) ν 0 { R ≥ i } . ν (34) 3 i ≥ n / 3 On the other hand, if j = s n ( x ) ≤ 2 n / 3, then T n − j ( x ) ∈ ∆ n − j and n − j ≥ n / 3. Using once more the T -invariance of ν and (32) � � � � � s n ≤ 2 n T − n + j (∆ n − j ) ν ≤ ν = ν (∆ n − j ) ≤ ν (∆ 0 ) ν 0 { R ≥ i } . 3 i ≥ n / 3 (35) 80
Finally, � ν { b n = k , r n = i , s n = j } 1 ≤ i < n / 3 2 n / 3 < j ≤ n � � � b j − i ◦ T i = k − 1 , r n = i , s n = j = ν 1 ≤ i < n / 3 2 n / 3 < j ≤ n � � � T − i { b j − i = k − 1 } ∩ T − i (∆ 0 ) ∩ T − j (∆ 0 ) ≤ ν 1 ≤ i < n / 3 2 n / 3 < j ≤ n � � � { b j − i = k − 1 } ∩ ∆ 0 ∩ T − j + i (∆ 0 ) = ν 1 ≤ i < n / 3 2 n / 3 < j ≤ n � � � { b ℓ = k − 1 } ∩ ∆ 0 ∩ T − ℓ (∆ 0 ) ≤ n ν ℓ> n / 3 � � � { R k = ℓ } ∩ ∆ 0 ∩ T − ℓ (∆ 0 ) = n ν (∆ 0 ) ν 0 ℓ> n / 3 � � R k > n ≤ n ν (∆ 0 ) ν 0 . (36) 3 81
Fom (33), (34), (35) and (36), we obtain for k ≥ 1 � � � R k ≥ n 2 . ν { b n = k } ≤ ν (∆ 0 ) ν 0 { R ≥ i } + n ν 0 (37) 3 i > n / 3 Now, observe that b n ( x ) = 0 means that T j ( x ) / ∈ ∆ 0 for all 1 ≤ j ≤ n . It follows from the T -invariance of ν and (32) that � � � ν { b n = 0 } = ν { R > j } = ν (∆ 0 ) ν 0 { R > j } ≤ ν (∆ 0 ) ν 0 { R ≥ j } . j ≥ n j ≥ n j > n / 3 Together with (31) and (37) this gives the conclusion. Corollary 3.13 For every H¨ older continuous ϕ : M → R and k ≥ 1 � � � � R ℓ > k T k − ϕ k � 1 ≤ C ϕ ◦ ˆ σ ℓ m γ 0 � ˆ m γ 0 { R ≥ k } + Ck . 3 ℓ ≥ 1 ℓ ≥ k / 3 This is a consequence of Lemma 3.11 and Proposition 3.12. 82
Partially hyperbolic attractors 83
SRB measures Consider a partially hyperbolic attractor T A M = E cu ⊕ E s with a trapping region U . We say that f is nonuniformly expanding along E cu on H ⊂ U if there is c > 0 and a Riemannian metric on M such that for all x ∈ H � n 1 log � Df − 1 | E cu f j ( x ) � < − c . lim inf (NUE) n n → + ∞ j =1 Theorem 4.1 (Alves, Dias, Luzzatto, and Pinheiro 2017) Assume A is a partially hyperbolic attractor with T A M = E s ⊕ E cu for which there exists H ⊂ U with m ( H ) > 0 on which f is NUE along E cu . Then there are ergodic SRB measures µ 1 , ..., µ ℓ on A such that m almost every x ∈ H belongs in the basin of µ j , for some 1 ≤ j ≤ ℓ . 84
Hyperbolic times Given 0 < σ < 1, we say that n ∈ N is a hyperbolic time for x if � n � Df − 1 | E cu f j ( x ) � ≤ σ k , for all 1 ≤ k ≤ n . j = n − k +1 As a consequence of Pliss Lemma, we have: Lemma 4.2 There exist σ > 0 and 0 < θ ≤ 1 such that if n − 1 � 1 log � Df − 1 | E cu f i ( x ) � < − c n i =0 then x has at least θ n hyperbolic times. Fix σ as in the lemma and define � � x ∈ H : n is a hyperbolic time for x H n = . (38) 85
cu -disks The dominated splitting T A M = E s ⊕ E cu gives rise to Df -invariant conefields { C s x } x ∈ U and { C cu x } x ∈ U in a neighborhood U of A . We say that D ⊂ U is a cu -disk if T x D ⊂ C cu for all x ∈ D . x The domination property implies that the forward iterates of cu -disks are still cu -disks. Assume that there is H ⊂ U with m ( H ) > 0 on which f is NUE along E cu . Then there exists some cu -disk D such that m D ( H ) > 0. We fix once and for all a cu -disk D in these conditions. 86
Lemma 4.3 There exist C , δ 1 > 0 such that for each x ∈ H n ∩ D there exists a neighborhood V n ( x ) of x in D so that: 1 f n maps V n ( x ) to a disk of radius δ 1 centred at f n ( x ); 2 for all 1 ≤ k ≤ n and y , z ∈ V n ( x ), dist f n − k ( V n ( x )) ( f n − k ( y ) , f n − k ( z )) ≤ σ k / 2 dist f n ( V n ( x )) ( f n ( y ) , f n ( z )); 3 for all y , z ∈ V n ( x ) log | det Df n | T y D | | det Df n | T z D | ≤ C dist f n ( D ) ( f n ( y ) , f n ( z )) . The sets V n ( x ) will be called hyperbolic pre-disks. 87
Local unstable manifolds Lemma 4.4 There are hyperbolic pre-disks V 1 , V 2 , · · · ⊂ D and integers n 1 < n 2 < · · · such that for B k = f n k ( V k ) m B k f n k ( H ∩ D ) lim = 1 . m B k ( B k ) k →∞ Considering an accumulation point of the sequence of the centers of the cu -disks ( B k ) k , by Ascoli-Arzel` a, there is an accumulation disk ∆ of ( B k ) k , which is necessarily an unstable disk. Moreover, by absolute continuity of the stable foliation, f is NUE for m ∆ almost every point in ∆. 88
Topological attractors Using the previous results we are able to decompose the attractor into a finite number of transitive pieces: Theorem 4.5 There are closed invariant sets Ω 1 , ..., Ω ℓ ⊂ A such that for Lebesgue almost every x ∈ H we have ω ( x ) = Ω j for some 1 ≤ j ≤ ℓ . Moreover, each Ω j is transitive and contains an unstable disk ∆ j on which f is NUE along E cu for m ∆ j almost every point in ∆ j . Fix any Ω := Ω j and ∆ := ∆ j ⊂ Ω j , with Ω j and ∆ j as in the theorem. Set m 0 = m ∆ . 89
Cylinders We fix a constant δ s > 0 so that stable disks W s δ s ( x ) are defined for all points x ∈ Ω. Given a subdisk ∆ ′ ⊂ ∆, define � C (∆ ′ ) = W s δ s ( x ) . x ∈ ∆ ′ Let π be the projection from C (∆ ′ ) onto ∆ ′ along stable disks. We say that a cu -disk γ u -crosses C (∆ ′ ) if π ( γ ′ ) = ∆ ′ for some connected component γ ′ of γ ∩ C (∆ ′ ). 90
Lemma 4.6 There are p ∈ ∆, δ > 0 and L ≥ 1 such that for each hyperbolic pre-disk V n ( x ) ⊂ ∆, there is 0 ≤ ℓ ≤ L such that f n + ℓ ( V n ( x )) u -crosses C ( B u δ 0 ( p )). Consider ∆ 0 = B u δ 0 ( p ). For each x ∈ H n there are 0 ≤ ℓ ≤ L and a cu -disk ω x n ⊂ V n ( x ) s.t. π ( f n + ℓ ( ω x ( ∗ ) n )) = ∆ 0 . 91
Inducing scheme We inductively define an m 0 mod 0 partition P of ∆ 0 such that for each ω ∈ P there is R = R ( ω ) ∈ N such that π ◦ f R maps ω bijectively to ∆ 0 . First step We fix some large n 0 ∈ N and choose a maximal set of points x 1 , . . . , x j n 0 ∈ ∆ 0 ∩ H n 0 such that the sets ω x i n 0 are pairwise disjoint and contained in ∆ 0 . Let x jn 0 P n 0 = { ω x 1 n 0 , . . . , ω n 0 } . These are the elements of the partition P constructed in the first step. For each 0 ≤ i ≤ j n 0 , we define the return time R | ω xi n 0 = n 0 + ℓ i , where 0 ≤ ℓ i ≤ L is the integer associated to ω x i n 0 as in ( ∗ ). 92
Let Z n 0 be the set of points z ∈ ∆ 0 ∩ H n 0 such that ω z n 0 could have been (but were not) chosen in the construction of P n 0 . For every z ∈ Z n 0 , the set ω z n 0 must either intersect some ω x i n 0 ∈ P n 0 or ∆ c 0 := ∆ \ ∆ 0 (for otherwise it would have been included in P n 0 ). 93
Satellites For each ω ∈ P n 0 ∪ { ∆ c 0 } we define � � Z ω x ∈ Z n 0 : ω x n 0 = n 0 ∩ ω � = ∅ and the n 0 -satellite associated to ω � S ω n 0 = V n 0 ( x ) . x ∈ Z ω n 0 Finally we consider � S ω S n 0 = n 0 . ω ∈P n 0 ∪{ ∆ c 0 } By construction we have � H n 0 ⊂ S n 0 ∪ ω. ω ∈P n 0 ∪{ ∆ c 0 } 94
Inductive step Assume now that the construction has been carried out up to time n − 1 for some n > n 0 : we have a collection of pairwise disjoint sets x jk P k = { ω x 1 k , ..., ω k } , for each n 0 ≤ k ≤ n − 1 , such that for any k � = k ′ ω ′ ∈ P k ′ ω ∩ ω ′ = ∅ . ω ∈ P k and ⇒ Define � ∆ n − 1 = ∆ 0 \ ω. ω ∈P n 0 ∪···∪P n − 1 95
We choose a maximal subset of points x 1 , . . . , x k n in H n such that the corresponding ω x i n are pairwise disjoint and contained in ∆ n − 1 . Define n , . . . , ω x jn P n = { ω x 1 n } For each 0 ≤ i ≤ j n , let 0 ≤ ℓ i ≤ L be as in ( ∗ ) and set R | ω n = n + ℓ i . xi Let Z n be the set of points z ∈ ∆ n − 1 ∩ H n such that ω z n could have been chosen in the construction of P n . Given ω ∈ P n 0 ∪ · · · ∪ P n ∪ { ∆ c 0 } , define Z ω n = { z ∈ Z n : ω z n ∩ ω � = ∅} and its n -satellite � S ω n = V n ( x ) . x ∈ Z ω n Finally, define � S ω S n = n . ω ∈P n 0 ∪···∪P n ∪{ ∆ c 0 } 96
By construction, we have for each n ≥ n 0 � H n ⊂ S n ∪ ω. ( ∗∗ ) ω ∈P n 0 ∪···∪P n ∪{ ∆ c 0 } Define � ∆ n = ∆ 0 \ ω. ω ∈P n 0 ∪···∪P n and � P = P n . n ≥ n 0 97
Estimates Lemma 4.7 There is C > 0 such that for all k ≥ n 0 , ω ∈ P k ∪ { ∆ c 0 } and n ≥ k n − k m 0 ( S ω 2 m 0 ( ω ) . n ) < C σ Lemma 4.8 ∞ � m 0 ( S n ) < ∞ . n = n 0 Thus, by Borel-Cantelli Lemma, m 0 almost every x ∈ ∆ 0 belongs in finitely many S n ’s. Since m 0 almost every x ∈ ∆ 0 has infinitely many hyperbolic times and ( ∗∗ ) holds, then m 0 almost every x ∈ ∆ 0 necessarily belongs in some element of P . 98
Integrability of the return time Let f R be the return map of a Young structure and F : γ 0 ∩ Λ → γ 0 ∩ Λ its quotient map. Set ∆ 0 = γ 0 ∩ Λ and m 0 = m γ 0 . Let ( H n ) n be as in (38). Note that there exists θ > 0 such that for m 0 almost every x ∈ ∆ 0 1 lim sup n # { 1 ≤ j ≤ n : x ∈ H j } > θ. (39) n →∞ Set R 0 = 0 and R k = � k − 1 j =0 R ◦ F j , for each k ≥ 1. We have ⇒ F i ( x ) ∈ H n − R i , x ∈ H n = (40) whenever R i ( x ) ≤ n < R i +1 ( x ), for some i ≥ 0. Proposition 4.9 Let F : ∆ 0 → ∆ 0 be a Gibbs-Markov map with respect to a partition P , and R : ∆ 0 → N constant in the elements of P . Assume there exist 1 a sequence ( H n ) n of sets in ∆ 0 for which (39) + (40) hold; 2 a sequence ( S n ) n of sets in ∆ 0 such that � n ≥ 1 m 0 ( S n ) < ∞ ; 3 L ∈ N such that H n ∩ { R > L + n } ⊂ S n , for all n ≥ 1 . Then, R is integrable with respect to m 0 . 99
Consider the ergodic F -invariant probability measure ν ≪ m 0 . It is enough to check the integrability of R with respect to ν . Assume by contradiction ∈ L 1 ( ν ). Since R is positive, Birkhoff’s Ergodic Theorem gives that R / � k − 1 � 1 R ( F i ( x )) → lim Rd ν = ∞ , (41) k k →∞ i =0 for ν almost every x ∈ ∆ 0 . Since � n ≥ 1 m 0 ( S n ) < ∞ , it follows from Borel-Cantelli Lemma that ν almost every x ∈ ∆ 0 belongs in a finite number of sets S n . Define s ( x ) = # { n ≥ 1 : x ∈ S n } for x ∈ ∆ 0 . We have for ν almost every x ∈ ∆ 0 � k − 1 � � 1 s ( F i ( x )) → sd ν = ν ( S n ) < ∞ . (42) k i =0 n ≥ 1 100
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