Gibbs-Markov-Young structures Jos e F. Alves International - - PowerPoint PPT Presentation

Gibbs-Markov-Young structures

Jos´ e F. Alves International Workshop on Differentiable Dynamical Systems 5-9 August, 2019 Jilin University, China

http://www.fc.up.pt/pessoas/jfalves/slides.pdf http://www.fc.up.pt/pessoas/jfalves/notes.pdf

1

SRB measures Physical measures Partially hyperbolic attractors Nonuniformly expanding maps

2

Expanding structures Gibbs-Markov maps Tower extensions Intermittent map

3

Hyperbolic product structures Young structures Tower extension Decay of correlations

4

Partially hyperbolic attractors Hyperbolic times and predisks Inducing scheme Decay of correlations

5

References

2

SRB measures

3

Physical measures

Let M be a compact Riemannian manifold. An invariant probability measure µ for f : M → M is a physical measure if, for a positive Lebesgue measure set of points x ∈ M, we have 1 n

n−1

• j=0

δf j(x)

w∗

− − − →

n→∞ µ,

(∗)

• r equivalently, for all continuous ϕ : M → R

lim

n→+∞

1 n

n−1

• j=0

ϕ(f j(x)) =

• ϕ dµ.

We define the basin of µ as B(µ) =

• x ∈ M : (∗) holds
• 4

Singular vs. absolutely continuous

The average of Dirac measures supported on an attracting periodic

• rbit is a physical measure. Moreover, the basin of this physical

measure contains the topological basin of the attracting periodic orbit. Any ergodic absolutely continuous invariant probability measure is a physical measure. Moreover, the basin of this physical measure contains almost every point of its support.

Exercise

Prove the statements above.

Hint for the second one: use that C 0(M) has a countable dense subset.

5

Bowen example: non-ergodic physical measure

Consider a flow with connexion of two saddles S and S′ as in the figure. Let −α, β > 0 be the eigenvalues of S and −α′, β′ be the eigenvalues of S.

R

S

S0

U

If the points in U \ {R} are attracted to the saddle connection, then necessarily αα′ ≥ ββ′. Now If αα′ > ββ′, then there is no physical measure in U. This is due to the fact that sojourn times on small neighborhoods of S and S′ are comparable with the whole previous time. If αα′ = ββ′ (and the return maps to neighborhoods of S and S′ have “nice” Taylor expansions), then there is a non-ergodic physical measure in U assigning positive weight to S and S′.

6

Doubling map

Consider f : S1 → S1 given by f (x) = 2x (mod 1). It is clear that f preserves the length of intervals, and so (...) f preserves the Lebesgue (length) measure m on the Borel sets.

Exercise

Show that m is ergodic.

Hint 1: Use Fourier series and the fact that f is ergodic iff for all ϕ ∈ L2(m) ϕ ◦ f = ϕ = ⇒ ϕ = const. Hint 2: Use that any interval becomes the whole interval after a finite number iterates, the fact that f preserves proportions and Lebesgue Density Theorem.

7

Solenoid attractor

Consider the unit disk D ⊂ C, the map F : S1 × D → S1 × D given by F(t, z) =

• 2t (mod 1), z

4 + 1 2e2πit

• ,

and the attractor A =

• n≥0

F n(S1 × D). Some facts about A:

1 A is a hyperbolic attractor. 2 Each x ∈ A has a stable leaf γs(x) and an unstable leaf γu(x). 3 The unstable leaves of points in A are contained in A.

8

Lifting

Considering π the projection in the first coordinate, we have a commuting diagram: A

F

− − → A π  

• 

 π S1

f

− − → S1 We lift the physical measure m (Lebesgue) of f to a measure µ on A in the following way: for any bounded φ : A → R, define φ± : S1 → R by φ+(x) = sup

ξ∈π−1(x)

φ(ξ) and φ−(x) = inf

ξ∈π−1(x) φ(ξ)

Lemma

lim

n→∞

• (φ ◦ F n)−dm = lim

n→∞

• (φ ◦ F n)+dm.

By Riesz-Markov Theorem, there is a probability measure µ supported

• n A such that for every continuous φ : A → R,
• φ dµ = lim

n→∞

• (φ ◦ F n)−dm = lim

n→∞

• (φ ◦ F n)+dm.

Moreover, µ is F-invariant and ergodic.

9

Some more facts: (1) A is foliated by unstable leaves. (2) The conditional measures of µ on unstable leaves are equivalent to the conditionals of Lebesgue measure on those leaves. (3) For any continuous φ : A → R and any ˜ ξ ∈ γs(ξ) we have lim

n→∞

1 n

n−1

• j=0

φ(F j(˜ ξ)) = lim

n→∞

1 n

n−1

• j=0

φ(F j(ξ)). (4) The stable foliation is absolutely continuous. µ ergodic supported on A ⇓ (1) µγu almost every point in an unstable leaf γu belongs in B(µ) ⇓ (2) mγu almost every point in an unstable leaf γu belongs in B(µ) ⇓ (3)+(4) µ is a physical measure

10

Lyapunov exponents

Let f : M → M be a diffeomorphism of a smooth manifold M. Given x ∈ M and v ∈ TxM, define λ(x, v) = lim

n→±∞

1 n log Df n(x)v, if these limits exist and coincide.

Theorem (Oseledec 1968)

Assume that f preserves an invariant probability measure µ. There exist measurable functions λi and a Df -invariant splitting TxM = ⊕iEi(x) with λ(x, v) = λi(x) for µ almost every x ∈ M and every v ∈ Ei(x). Moreover, if µ is ergodic, then λi and dim(Ei) are constant µ almost everywhere. Each λi is called a Lyapunov exponent of f (with respect to µ). We define the regular set R ⊂ M as the set of points for which the Lyapunov exponents are defined.

11

SRB measures

Theorem (Pesin 1976)

If x ∈ R has at least one positive Lyapunov exponent, then there is a small disk γu(x) ⊂ M tangent to ⊕λi>0Ei(x) such that for all y ∈ γu(x) lim sup

n→∞

1 n log d(f −n(y), f −n(x)) < 0. γu(x) is called the local unstable manifold of x ∈ R. A local stable manifold γs(x) can be obtained similarly for a point x ∈ R with at least one negative Lyapunov exponent. The measure µ is called an Sinai-Ruelle-Bowen (SRB) measure if it has at least one positive Lyapunov exponent and the conditionals {µγu} of the Rokhlin decomposition of µ on local unstable manifolds are absolutely continuous with respect to the Lebesgue conditionals {mγu}.

12

Absolute continuity

Given D, D′ ⊂ M embedded disks intersecting transversally a set {γs(x)}x of stable leaves, define the holonomy map h :

xγs(x) ∩ D −

→

xγs(x) ∩ D′

assigning to z ∈ γs(x) ∩ D the unique point in γs(x) ∩ D′. The stable foliation is absolutely continuous if for any A ⊂

x γs(x) ∩ D, we have

mD(A) > 0 iff mD′(h(A)) > 0.

Theorem (Pesin 1976)

Let f : M → M be a C 2 diffeomorphism having all Lyapunov exponents nonzero with respect to an ergodic invariant probability measure µ. Then the stable foliation is absolutely continuous.

Corollary

Every ergodic SRB measure with non-zero Lyapunov exponents is a physical measure.

13

Attractors

Let f : M → M be a C 2 diffeomorphism of a compact connected Riemannian manifold. Let m denote Lebesgue measure on M. Given a submanifold γ ⊂ M, let mγ denote the Lebesgue measure on γ. Let A ⊂ M be a compact and f -invariant set: f (A) = A. A is called an attractor if there exists an open neighborhood U of A with f

• U
• ⊂ U such that

A =

• n≥0

f n(U). U is called a trapping region for the attractor A.

14

Partially hyperbolic attractors

An attractor A is partially hyperbolic if there is a Df -invariant splitting TAM = E cs ⊕ E cu such that for some choice of a Riemannian metric on M E cs ⊕ E cu is a dominated splitting: there is 0 < λ < 1 such that Df | E cs

x · Df −1 | E cu f (x) ≤ λ

∀x ∈ A. E cs is uniformly contracting or E cu is uniformly expanding: there is 0 < λ < 1 such that Df | E cs

x ≤ λ

∀x ∈ A

• r

Df −1 | E cu

f (x)−1 ≤ λ

∀x ∈ A. To emphasize the uniform behavior, we will write E s instead of E cs in the first case, and E u instead of E cu in the second one.

15

Hyperbolic case

Assume that A is a hyperbolic attractor, i.e. TAM = E s ⊕ E u, with E s uniformly contracting and E u uniformly expanding. Classical results by Sinai, Ruelle and Bowen give the existence of SRB measures and good statistical properties for these measures for: Anosov diffeomorphisms (Sinai 1972; Bowen 1975) Axiom A attractors for diffeomorphisms (Ruelle 1976) Axiom A attractors for flows (Bowen and Ruelle 1975) These results were proved using a codification of the system, via finite Markov partitions, and symbolic dynamics (subshifts of finite type). The SRB measures are actually particular cases of equilibrium states. Not so easy to implement in nonuniformly hyperbolic dynamics (Sarig 1999; Sarig 2003; Buzzi and Sarig 2003; Sarig 2013; Ledrappier, Lima, and Sarig 2016).

16

Case E cs ⊕ E u

Theorem (Pesin and Sinai 1982)

Let A ⊂ M be a compact attractor on which f is partially hyperbolic with splitting TAM = E cs ⊕ E u. Then there are SRB measures supported on A. This result was proved considering a local unstable manifold γ ⊂ A and showing that any weak* accumulation point of the sequence µn = 1 n

n−1

• j=0

f j

∗mγ

is an SRB measure.

17

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: λc

+(x) = lim sup n→∞

1 n log Df n|E cs

x .

Theorem (Bonatti and Viana 2000)

Let A ⊂ M be an attractor on which f is partially hyperbolic with splitting TAM = E cs ⊕ E u. Assume that for any unstable manifold γu we have λc

+(x) < 0

for a positive mγu measure set of points x ∈ γu. Then there are ergodic SRB measures µ1, ..., µℓ supported on A such that for m almost every x with ω(x) ⊂ A we have x ∈ B(µj) 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 TAM = 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 lim inf

n→+∞

1 n

n

• j=1

log Df −1|E cu

f j (x) < −c.

(NUE) 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 lim sup

n→+∞

1 n

n

• j=1

log Df −1|E cu

f j (x) < −c.

(SNUE)

19

Theorem (Alves, Bonatti, and Viana 2000)

Let A =

n≥0 f n(U) be a partially hyperbolic set with TAM = 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 µ1, ..., µℓ on A such that for Lebesgue almost every x ∈ H we have x ∈ B(µj) 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 H1, H2, H3 · · · ⊂ D such that the weak* accumulation points of µn = 1 n

n−1

• j=0

f j

∗(mD|Hj)

have the “SRB property”. Moreover, there is some α > 0 such that µn(M) = 1 n

n−1

• j=0

mD(Hj) ≥ α, for large n. Difficulty: weak* accumulation points of (µn) 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 TAM = 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 µ1, ..., µℓ on A such that for Lebesgue almost every x ∈ H we have x ∈ B(µj) 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

lim

n→+∞

1 n

n

• j=1

log Df −1|E cu

f j (x) < 0.

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: lim sup

n→∞

1 n log Df n(x)v > 0, ∀v ∈ E cu \ {0}. (∗) 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 Corµ(ϕ, ψ ◦ f n) =

• ϕ(ψ ◦ f n)dµ −
• ϕdµ
• ψdµ
• .

Taking ϕ and ψ the characteristic functions of Borel sets, we obtain the usual notion of mixing when Corµ(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¨

• lder continuous: there is η > 0 such that

sup

x=y

|ϕ(x − ϕ(y)| dist(x, y)η < ∞. Consider H = {ϕ : M → R| ϕ is H¨

• lder 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 TAM = E cs ⊕ E u and the leaves of Fu 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¨

• lder 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¨

• lder 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¨

• lder 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 TAM = 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 E(x) = min

• N ≥ 1: 1

n

n−1

• i=0

log Df −1 | E cu

f i(x) < −c,

∀ n ≥ N

• Theorem (Alves and Pinheiro 2010)

If there is a local unstable disk D ⊂ A such that mD{E > n} n−α for some α > 0, then Corµ(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−cnθ for some c > 0 and 0 < θ ≤ 1, then there is c′ > 0 such that Corµ(ϕ, ψ ◦ f n) e−c′nθ 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 = Rd/Zd and f : T d → T d given by the quotient of a linear map having a diagonal matrix with integer eigenvalues λ1, . . . , λd ≥ 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 lim inf

n→+∞

1 n

n

• j=1

log Df (f j(x))−1 < −c, (NUE) 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 E(x) = min

• N ≥ 1: 1

n

n−1

• i=0

log Df (f i(x))−1 < −c, ∀ n ≥ N

• 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 Corµ(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−cnθ for some c > 0 and 0 < θ ≤ 1, then there is c′ > 0 such that Corµ(ϕ, ψ ◦ f n) e−c′nθ 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 lim inf

n→∞

1 n#

• 0 ≤ j < n : f j(x) /

∈ B

• ≥ θ.

It follows that for small δ > 0 there is c > 0 such that for Lebesgue almost every x ∈ M lim sup

n→+∞

1 n

n−1

• j=0

log Df (f j(x))−1 ≤ −c . 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: ∃ JF > 0 such that for each A ⊂ ω ∈ P

m(F(A)) =

• A

JFdm.

3 Separation: for all x, y ∈ ∆0 there is

s(x, y) = min

• n ≥ 0 : F n(x), F n(y) lie in distinct elements of P
• .

4 Bounded distortion: ∃ K > 0 and 0 < β < 1 s.t. for all x, y ∈ ω ∈ P

log JF(x) JF(y) ≤ Kβs(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 log

• det DF(x)

det DF(y)

• ≤ K dist(F(x), F(y))γ.

Show that F is a Gibbs-Markov map.

33

Reference spaces

Consider the space Fβ(∆0) =

• ϕ : ∆0 → R s.t. |ϕ|β ≡ sup

x=y

|ϕ(x) − ϕ(y)| βs(x,y) < ∞

• .

endowed with the norm |ϕ|β + ϕ∞, and F+

β (∆0) =

• ϕ ∈ Fβ(∆0) : ϕ ≥ c for some c > 0
• .

Exercise

Show that Fβ(∆0) is relatively compact in L1(∆0). An F-invariant probability measure is exact (⇒ mixing ⇒ ergodic) if A ∈

n≥0F −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, dν/dm belongs in Fβ(∆0) and there is K > 0 such that 1 K ≤ dν dm ≤ K. The idea is to prove that the sequence of densities of the measures µn = 1 n

n−1

• j=0

F j

∗m.

is bounded in Fβ(∆0), and so it has an accumulation point in L1(∆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 ∈ L1(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¨

• lder continuous functions from M to R.

Theorem (Young 1999)

Assume that f has an induced Gibbs-Markov map f R with R ∈ L1(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 Corµ(ϕ, ψ ◦ f n) n−α+1; 2 if m{R > n} e−cnθ for some c > 0 and 0 < θ ≤ 1, then

Corµ(ϕ, ψ ◦ f n) e−c′nθ 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 T(x, ℓ) = (x, ℓ + 1), if ℓ < R(x) − 1; (f R(x), 0), if ℓ = R(x) − 1. The map π: ∆ − → M (x, ℓ) − → f ℓ(x) (1) satisfies f ◦ π = π ◦ T.

38

The ℓth level of the tower is the set ∆ℓ = {(x, ℓ) ∈ ∆}. The 0th level is naturally identified with the set ∆0 ⊂ M. Under this identification we have T R = f R : ∆0 − → ∆0 Gibbs-Markov. The ℓth 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 x0, y0 ∈ ∆0 such that x = T ℓ(x0) and y = T ℓ(y0). Set s(x, y) = s(x0, y0). Define s(x, y) = 0 for all other points x, y ∈ ∆. We consider as before Fβ(∆) =

• ϕ : ∆ → R | ∃C > 0 : |ϕ(x) − ϕ(y)| ≤ Cβs(x,y), ∀x, y ∈ ∆
• .

39

Define for ϕ ∈ Fβ Cϕ = sup

x=y

|ϕ(x) − ϕ(y)| βs(x,y) . (2) We also consider F+

β (∆) = {ϕ ∈ Fβ(∆) | ∃c > 0 : ϕ ≥ c} .

If ϕ ∈ F+

β (∆), then 1/ϕ is bounded. Given ϕ ∈ F+ β (∆), set

C +

ϕ = max

• Cϕ, ϕ∞,
• 1

ϕ

• ∞
• .

(3)

Theorem 2.3

If R ∈ L1(m), then the tower map T : ∆ → ∆ has a unique ergodic invariant probability measure ν which is equivalent to m. Moreover, dν/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 ∈ L1(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 Corν(ϕ ◦ π, ψ ◦ π ◦ T n) = Corµ(ϕ, ψ ◦ f n) for all ϕ, ψ and n ≥ 1. 3 given ϕ ∈ H, there is some β > 0 such that ϕ ◦ π ∈ Fβ(∆).

The proof of Young Theorem is then a consequence of:

Theorem 2.4

Assume that gcd{R} = 1. For all ϕ ∈ Fβ(∆) and all ψ ∈ L∞(m)

1 if m{R > n} n−α for some α > 0, then Corν(ϕ, ψ ◦ T n) n−α+1; 2 if m{R > n} e−cnθ for some c > 0 and 0 < θ ≤ 1, then

Corν(ϕ, ψ ◦ T n) e−c′nθ for some c′ > 0.

41

Given ϕ ∈ L∞(m) with ϕ = 0, define ϕ∗ = 1

• (ϕ + 2ϕ∞)dν (ϕ + 2ϕ∞).

(4) Note that ϕ∗ is strictly positive and its integral with respect to ν is 1.

Lemma 2.5

For all ϕ ∈ Fβ(∆) with ϕ = 0 we have

1 ϕ∗ ∈ F+

β (∆) and 1/3 ≤ ϕ∗ ≤ 3;

2 Corν(ϕ, ψ ◦ T n) ≤ 3ϕ∞ψ∞|T n

∗ λ − ν| for all ψ ∈ L∞(m),

where λ is the probability measure on ∆ such that dλ/dν = ϕ∗. We have ϕ∞ ≤ ϕ + 2ϕ∞ ≤ 3ϕ∞. (5) Since ν is a probability measure, we get 1 3ϕ∞ ≤ 1

• (ϕ + 2ϕ∞)dν ≤

1 ϕ∞ . (6)

42

For all x, y ∈ ∆ we have ϕ∗(x) − ϕ∗(y) βs(x,y) = 1

• (ϕ + 2ϕ∞)dν · ϕ(x) − ϕ(y)

βs(x,y) . (7) Since ϕ ∈ Fβ(∆), then ϕ∗ ∈ Fβ(∆). From (5) and (6) we get 1/3 ≤ ϕ∗ ≤ 3, and so ϕ∗ ∈ F+

β (∆). Set a =

• (ϕ + 2ϕ∞)dν. We have

Corµ(ϕ, ψ ◦ T n) =

• ϕ(ψ ◦ T n)dν −
• ϕdν
• ψdν
• = a
• ϕ∗(ψ ◦ T n) dν −
• ϕ∗dν
• ψ dν
• = a
• (ψ ◦ T n) dλ −
• ψ dν
• = a
• ψ dT n

∗ λ −

• ψ dν
• ≤ aψ∞|T n

∗ λ − ν|.

Observing that a ≤ 3ϕ∞, 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 = dλ/dm belongs in F+

β (∆) we have:

1 if m{R > n} ≤ Cn−ζ for some C > 0 and ζ > 1, then

|T n

∗ λ − ν| ≤ C ′n−ζ+1 for some C ′ > 0;

2 if m{R > n} ≤ Ce−cnη for some C, c > 0 and 0 < η ≤ 1, then

|T n

∗ λ − ν| ≤ C ′e−c′nη for some C ′, c′ > 0;

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 = dλ dm and ϕ′ = dλ′ 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

• nto a union of elements of Q × Q. For each n ≥ 1, let

(Q × Q)n :=

n−1

• i=0

(T × T)−i(Q × Q), 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 dν/dm is bounded, we may find n0 ∈ N and γ0 > 0 such that m(T −n(∆0) ∩ ∆0) ≥ γ0, ∀n ≥ n0. Consider R : ∆ → Z defined as

• R(x) = min{n ≥ 0 : T n(x) ∈ ∆0}.

45

We introduce a sequence of times 0 ≡ τ0 < τ1 < τ2 < ... in ∆ × ∆ by τ1(x, x′) = n0 + R(T n0(x)), τ2(x, x′) = τ1 + n0 + R(T τ1+n0(x′)), τ3(x, x′) = τ2 + n0 + R(T τ2+n0(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

• τi : (T τi(x), T τi(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 ≥ 2n0 and if S(x, x′) = n, then S|(Q×Q)n(x,x′) = n and (T × T)n((Q × Q)n(x, x′)) = ∆0 × ∆0.

46

A simplified model

Assume first that JT and the densities dλ/dm and dλ′/dm are constant

• n each element of the partition. We may write

|T n

∗ λ − T n ∗ λ′|

=|π∗(T × T)n

∗P − π′ ∗(T × T)n ∗P|

≤|π∗(T × T)n

∗(P|{S > n}) − π′ ∗(T × T)n ∗(P|{S > n})|

+

n

• i=1

|π∗(T × T)n

∗(P|{S = i}) − π′ ∗(T × T)n ∗(P|{S = i})|

=

• π∗(T × T)n

∗(P|{S > n}) − π′ ∗(T × T)n ∗(P|{S > n})

• +

n

• i=1

|T n−i

∗

• π∗(T × T)i

∗(P|{S = i}) − π′ ∗(T × T)i ∗(P|{S = i})

• |

In this last equality we have used that T n−i ◦ π = π ◦ (T × T)i and T n−i ◦ π′ = π′ ◦ (T × T)i.

47

Now, if S(x, x′) = i, then π∗(T × T)i

∗

• P|(Q × Q)i(x, x′)
• = 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′| ≤ 2P{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

• T n

∗ λ − T n ∗ λ′

≤ 2P{S > n} + K

∞

• i=1

θi(i + 1)P

• S >

n 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−cnη for some c > 0 and 0 < η ≤ 1, then

P{S > n} e−c′nη 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 + x1+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

Corµ(ϕ, ψ ◦ f n) 1/n1/γ−1;

3 if γ ≥ 1, then δ0 is the unique physical measure of f .

50

Consider (xn)n the sequence in [0, 1/2] defined recursively as x1 = 1/2 and f (xn+1) = xn, ∀n ≥ 1. Letting J0 = [1/2, 1] and Jn = [xn+1, xn], ∀n ≥ 1, consider the (m mod 0) partition of I P = {Jn : n ≥ 0}. Defininig R|Jn = 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 ∈ Jn;

3 m{R > n} ≈ 1/n1/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/n1/γ. For γ < 1, we have R ∈ L1(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 µ =

∞

• j=0

f j

∗(ν|{R > j}).

an ergodic absolutely continuous f -invariant measure. Not finite!

Proposition 2.10

If γ ≥ 1, then δ0 is the unique physical measure of f . First of all observe that for all n ≥ 1 we have µ(Jn) = ν  

k≥n

Jk   = ν{R > n} ≈ m{R > n} ≈ n−1/γ. It follows that dµ dm|Jn ≈ 1 m(Jn)n−1/γ ≈ n. (8)

52

We are going to show that for m almost every x ∈ I we have 1 n

n−1

• j=0

δf j(x)

w∗

− → δ0, It is enough to show that for large N and small ε > 0 we have lim

n→∞

1 n#

• 0 ≤ j < n : f j(x) ∈ [0, xN)
• > 1 − ε.

Using (8), we may find N1 ≫ N such that µ([xN, 1]) µ([xN1, 1]) < ε. Consider F the first return map to [xN1, 1].

Exercise

The measure η = 1 µ([xN1, 1])µ|[xN1, 1] is an F-invariant ergodic probability measure equivalent to m|[xN1, 1].

53

Hence, Birkhoff Ergodic Theorem gives that for m almost all x ∈ [xN1, 1] lim

n→∞

1 n#

• 0 ≤ j < n : F j(x) ∈ [0, xN)
• = η([0, xN)).

We have η([0, xN)) = 1 − η([xN, 1]) = 1 − µ([xN, 1]) µ([xN1, 1]) > 1 − ε. Observing that the fraction of time spent in (0, xN) 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 Γs = {γs} of stable disks and a family Γu = {γu} of unstable disks such that

• Λ = (∪γu) ∩ (∪γs);
• dim γu + dim γs = dim M;
• each γs and γu 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′ ∈ Γu, define Θγ,γ′ : γ ∩ Λ → γ′ ∩ Λ by Θγ,γ′(x) = γs(x) ∩ γ, (9) and Θγ : Λ → γ ∩ Λ by Θγ(x) = Θγu(x),γ(x). (10) We say that the hyperbolic product structure is measurable if the maps Θγ,γ′ and Θγ are measurable, for all γ, γ ∈ Γu. Λ0 ⊂ Λ is called an s-subset if Λ0 = Γs

0 ∩ Γu for some Γs 0 ⊂ Γs.

Λ0 ⊂ Λ is called a u-subset if Λ0 = Γu

0 ∩ Γs for some Γu 0 ⊂ Γu. 57

A set Λ with a measurable product structure for which (Y1)-(Y5) below hold will be called a Young structure. (Y1) Markov: ∃ pairwise disjoint s-subsets Λ1, Λ2, · · · ⊂ Λ such that

◮ mγ(Λ ∩ γ) > 0 and mγ(Λ \ ∪iΛi) ∩ γ) = 0 for all γ ∈ Γu; ◮ ∀i ∈ N ∃Ri ∈ N such that f Ri(Λi) is a u-subset and for all x ∈ Λi

f Ri(γs(x)) ⊂ γs(f Ri(x)) and f Ri(γu(x)) ⊃ γu(f Ri(x)).

We define the recurrence time R : Λ → N and the return map f R : Λ → Λ R|Λi = Ri and f R|Λi = f Ri. 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 Λi’s.

58

Let C > 0 and 0 < β < 1 be constants depending only on f and Λ . (Y2) Contraction on stable disks: for all γ ∈ Γs 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).

(Y3) Expansion on unstable disks: for all γ ∈ Γu, all Λi and x, y ∈ γ ∩ Λi

◮ 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).

(Y4) Absolute continuity of Γs: for all γ, γ′ ∈ Γu, the map Θγ,γ′ is absolutely continuous; moreover, letting ξγ,γ′ denote the density of (Θγ,γ′)∗mγ with respect to mγ′, we have for all x, y ∈ γ′ ∩ Λ 1 C ≤ ξγ,γ′(x) ≤ C and log ξγ,γ′(x) ξγ,γ′(y) ≤ Cβs(x,y). (Y5) Bounded distortion: ∃γ0 ∈ Γu such that for all Λi and x, y ∈ γ0 ∩ Λi log det Df R|Txγ0 det Df R|Tyγ0 ≤ Cβs(f R(x),f R(y)).

59

We say that the Young structure has integrable return times if for some (and hence for all) γ ∈ Γu, we have

• γ∩Λ

Rdmγ < ∞. Let H be the space of H¨

• lder 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 mγ{R > n} ≤ Cn−a for some γ ∈ Γu and C > 0, a > 1, then for all

ϕ, ψ ∈ H there exists C ′ > 0 such that Corµ(ϕ, ψ ◦ f n) ≤ C ′n−a+1;

2 if mγ{R > n} ≤ Ce−cna for some γ ∈ Γu and constants C, c > 0 and

0 < a ≤ 1, then for all ϕ, ψ ∈ H there exists C ′ > 0 such that Corµ(ϕ, ψ ◦ f n) ≤ C ′e−c′na. Contribution of Gou¨ ezel, Korepanov, Kosloff, and Melbourne 2019 for the simplified set of assumptions.

60

H¨

• lder against bounded

In the non-invertible case we have Cor(ϕ, ψ ◦ f n) ≤ Cϕ,ψan with an → 0 as n → ∞, for ϕ ∈ H and ψ ∈ L∞(m). We have for some K > 0 Cϕ,ψ = Kϕηψ∞.

Lemma 3.2

If f : M → M is a diffeomorphism, then Corµ(ϕ, ψ ◦ f n) → 0 does not hold for all ϕ ∈ H and all ψ ∈ L∞. Take ϕ ∈ H with

• ϕdµ = 0 and ϕ1 = 0. We may write for n ≥ 1

ϕ1 = ϕ ◦ f −n1 = sup

ψ∞=1

• (ϕ ◦ f −n)ψdµ

= sup

ψ∞=1

• ϕ(ψ ◦ f n)dµ

= sup

ψ∞=1

Corµ(ϕ, ψ ◦ f n) 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 νn = 1 n

n−1

• j=0

(f R)j

∗mγu,

some γ ∈ Γu.

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. µ = 1 ∞

j=0 ν{R > j} ∞

• j=0

f j

∗(ν|{R > j}).

(11)

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 ˆ T(x, ℓ) = (x, ℓ + 1), if ℓ + 1 < R(x); (f R(x), 0), if ℓ + 1 = R(x). The ℓ-level of the tower is ˆ ∆ℓ = {(x, ℓ) ∈ 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 = Λi. 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

• f ˆ

∆.

63

Setting π : ˆ ∆ − → M (x, ℓ) − → f ℓ(x) 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 ˆ ν = 1 ∞

j=0 ν{R > j} ∞

• j=0

ˆ T j

∗(ν|{R > j})

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 γ0 ∈ Γu as in (Y5), we define the quotient map of f R on γ0 ∩ Λ F : γ0 ∩ Λ − → γ0 ∩ Λ x − → Θγ,γ0 ◦ f R(x), where γ = γu(f R(x)).

Proposition 3.6

F is Gibbs-Markov with respect to the mγ0 mod 0 partition P = {γ0 ∩ Λ1, γ0 ∩ Λ2, . . . } of γ0 ∩ Λ.

Lemma 3.7

Let F : γ0 ∩ Λ → γ0 ∩ Λ be the quotient map of f R : Λ → Λ. If ν is an SRB measure of f R, then ν0 = (Θγ0)∗ν is the F-invariant probability measure such that ν0 ≪ mγ0.

65

Quotient tower

Fix γ0 ∈ Γu as in (Y5), and the quotient map F : γ0 ∩ Λ → γ0 ∩ Λ. Consider the tower map T : ∆ → ∆ of F with recurrence time R. Notice that for all i ≥ 1 R|γ0∩Λi = R|Λi = Ri. Since γ0 ∩ Λ ⊂ Λ, it easily follows that for all ℓ ≥ 0 we have ∆ℓ ⊂ ˆ ∆ℓ and T = ˆ T|∆. (12) Moreover, ˆ T ◦ Θ = Θ ◦ T, where Θ : ˆ ∆ − → ∆ (x, ℓ) − → (Θγ0(x), ℓ) . (13)

Proposition 3.8

If ˆ ν is the ergodic SRB measure of ˆ T, then Θ∗ˆ ν is the unique ergodic T-invariant probability measure absolutely continuous with respect to mγ0.

66

Decay of correlations

We have π ◦ ˆ T = f ◦ π and Θ ◦ ˆ T = T ◦ Θ. (14) Let ˆ ν be the unique ergodic SRB measure of 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 ≪ mγ0. 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 ˆ ϕ = ϕ ◦ π. (16)

Exercise

Corµ(ϕ, ψ ◦ f n) = Corˆ

ν( ˆ

ϕ, ˆ ψ ◦ ˆ T n). For proving Theorem 3.1, it is enough to obtain the estimates for Corν( ˆ ϕ, ˆ ψ ◦ ˆ T n). The idea is to reduce it to a problem on the quotient tower T : ∆ → ∆, and apply Theorem 2.6.

67

Given k ≥ 1, define ˆ Qk =

k−1

• j=0

ˆ T −j ˆ Q. (17) Define the discretisation ϕk : ˆ ∆ → R of ˆ ϕ, setting for each Q ∈ ˆ Q2k ϕk|Q = inf{ ˆ ϕ ◦ ˆ T k(x): x ∈ Q}. (18) ϕk may as well be thought of as function on ∆.

Proposition 3.9

For all ϕ, ψ ∈ H and 1 ≤ k ≤ n, Corˆ

ν( ˆ

ϕ, ˆ ψ ◦ ˆ T n) ≤ Corν(ϕk, ψk ◦ T n) + 2ϕ0 ˆ ψ ◦ ˆ T k − ψk1 + 2ψ0 ˆ ϕ ◦ ˆ T k − ϕk1. 1 is the L1-norm with respect to the probability measure ˆ ν on ˆ ∆. Since Θ∗ˆ ν = ν, it coincides with the L1-norm with respect to the probability measure ν on ∆ for functions which are constant on stable disks.

68

Step 1

Since the probability measure ˆ ν is ˆ T-invariant, we may write Corˆ

ν( ˆ

ϕ, ˆ ψ ◦ ˆ T n) =

• ( ˆ

ψ ◦ ˆ T n) ˆ ϕd ˆ ν −

• ˆ

ψd ˆ ν

• ˆ

ϕd ˆ ν

• ≤
• ψk ◦ ˆ

T n−k ˆ ϕd ˆ ν −

• ψkd ˆ

ν

• ˆ

ϕd ˆ ν

• +
• ( ˆ

ψ ◦ ˆ T k − ψk) ◦ ˆ T n−k ˆ ϕd ˆ ν

• +
• (ψk − ˆ

ψ ◦ ˆ T k)d ˆ ν

• ˆ

ϕd ˆ ν

• ≤ Corˆ

ν( ˆ

ϕ, ψk ◦ ˆ T n−k) + 2 ˆ ψ ◦ ˆ T k − ψk1 ϕ0. (19)

69

Step 2

Note that as ˆ ν is ˆ T-invariant, then ˆ T∗ preserves absolute continuity with respect to ˆ ν. Defining ζk as the density of the signed measure ˆ T k

∗ (ϕk ˆ

ν) with respect to ˆ ν, Corˆ

ν( ˆ

ϕ, ψk ◦ ˆ T n−k) =

• ψk ◦ ˆ

T n−k ˆ ϕd ˆ ν −

• ψkd ˆ

ν

• ˆ

ϕd ˆ ν

• ≤
• ψk ◦ ˆ

T n−kζkd ˆ ν −

• ψkd ˆ

ν

• ζkd ˆ

ν

• +
• (ψk ◦ ˆ

T n−k)( ˆ ϕ − ζk)d ˆ ν

• +
• ψkd ˆ

ν

• ( ˆ

ϕ − ζk)d ˆ ν

• ≤ Corˆ

ν(ζk, ψk ◦ ˆ

T n−k) + 2ψ0

• ( ˆ

ϕ − ζk)d ˆ ν

• .

(20)

70

Observing that d ˆ T k

∗ (( ˆ

ϕ ◦ ˆ T k)ˆ ν) d ˆ ν = ˆ ϕ and d ˆ T k

∗ (ϕk ˆ

ν) d ˆ ν = ζk, we obtain

• ( ˆ

ϕ − ζk)d ˆ ν

• =
• d ˆ

T k

∗ (( ˆ

ϕ ◦ ˆ T k)ˆ ν) −

• d ˆ

T k

∗ (ϕk ˆ

ν)

• ≤
• | ˆ

ϕ ◦ ˆ T k − ϕk|d ˆ ν, which together with (20) gives Corˆ

ν( ˆ

ϕ, ψk ◦ ˆ T n−k) ≤ Corˆ

ν(ζk, ψk ◦ ˆ

T n−k) + 2ψ0 ˆ ϕ ◦ ˆ T k − ϕk1. Recalling (19), it just remains to show that Corˆ

ν(ζk, ψk ◦ ˆ

T n−k) = Corν(ϕk, ψk ◦ T n). (21)

71

Step 3

Observe that

• (ψk ◦ ˆ

T n−k)ζkd ˆ ν =

• ψkd( ˆ

T n−k

∗

(ζk ˆ ν)) =

• ψkd( ˆ

T n

∗ (ϕk ˆ

ν)). Since ψk and ϕk are constant on stable disks and (14), (15) hold

• ψkd( ˆ

T n

∗ (ϕk ˆ

ν)) =

• ψkd(Θ∗ ˆ

T n

∗ (ϕk ˆ

ν)) =

• ψkd(T n

∗ (ϕkν)) =

• (ψk◦T n)ϕkdν.

It follows that

• (ψk ◦ ˆ

T n−k)ζkd ˆ ν =

• (ψk ◦ T n)ϕkdν.

On the other hand,

• ψkd ˆ

ν

• ζkd ˆ

ν =

• ψkdν
• d( ˆ

T k

∗ (ϕk ˆ

ν)) =

• ψkdν
• ϕkdν.

The last two formulas yield (21).

72

Let us estimate Corν(ϕk, ψk ◦ T n). Consider ϕ∗

k as in (4). Let λ∗ k be such

that dλ∗

k/dν = ϕ∗

• k. Lemma 2.5 gives

Corν(ϕk, ψk ◦ T n) ≤ 3ϕ0ψ0|T n

∗ λ∗ k − ν|.

(22) Setting λk = T 2k

∗ λ∗ k and ρ = dν/dmγ0,

φk := dλk dmγ0 = dT 2k

∗ λ∗ k

dmγ0 and dλ∗

k

dmγ0 = ϕ∗

kρ.

(23) Take k ≈ n/4. Since T n

∗ λ∗ k = T n−2k ∗

λk, it follows from (22) that Corν(ϕk, ψk ◦ T n) ≤ 3ϕ0ψ0|T n−2k

∗

λk − ν|. (24)

Lemma 3.10

There exists C > 0 such that C +

φk ≤ C, for all k ≥ 1.

We apply Theorem 2.6 to get the desired conclusions on the decay of |T n−2k

∗

λk − ν|, and so on the term Corν(ϕk, ψk ◦ T n) in Proposition 3.9, by (22). Note that n − 2k ≈ n/2.

73

We are left to estimate the L1-norms in Proposition 3.9. Define for x ∈ ˆ ∆ and k ≥ 1 bk(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 bk is constant on stable disks.

Lemma 3.11

For every H¨

• lder continuous ϕ : M → R there are C > 0 and 0 < σ < 1

such that for all k ≥ 1 and x ∈ ∆ we have | ˆ ϕ ◦ ˆ T k(x) − ϕk(x)| ≤ C

• σbk(x) + σbk( ˆ

T k(x))

.

74

Take Q ∈ ˆ Q2k such that x ∈ Q. It follows from (16) and (18) that there exists y ∈ Q such that for some η > 0 | ˆ ϕ ◦ ˆ T k(x) − ϕk(x)| = |ϕ ◦ π ◦ ˆ T k(x) − ϕ ◦ π ◦ ˆ T k(y)| ≤ |ϕ|η dist(π( ˆ T k(x)), π( ˆ T k(y)))η. (25) Note that as x, y belong in Q ∈ ˆ Q2k, then b2k(y) = b2k(x). In particular, we have bk(y) = bk(x). Take z ∈ Q such that z ∈ γu(x) ∩ γs(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 y0 = ˆ T −ℓ(y) and z0 = 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(z0)), π( ˆ T k+ℓ(y0)))) = dist(f k+ℓ(z0), f k+ℓ(y0)). (27) Set b = bk+ℓ(y0) = bk(x) ≥ 0 and Rb = b−1

j=0 R((f R)j(y0)). Note that

k + ℓ = Rb + j, with j < R(f Rb(y0)). By (Y2), dist(f k+ℓ(z0), f k+ℓ(y0)) = dist(f j(f Rb(z0)), f j(f Rb(y0))) ≤ C dist(f Rb(z0), f Rb(y0)) = C dist((f R)b(z0), (f R)b(y0)). (28) Recalling that b = bk(x), it follows from (27), (28) and (Y2) that dist(π( ˆ T k(z)), π( ˆ T k(y))) ≤ Cβbk(x) dist(z0, y0). (29) We are left to estimate the other term on the right hand side of (26).

76

Since ˆ T k(Q) ∈ ˆ Qk, we have ˆ T k(x), ˆ T k(z) in the same unstable disk of some Q′ ∈ ˆ

• Qk. This implies that

bk( ˆ T k(x)) = bk( ˆ T k(z)) ≤ s( ˆ T k(x), ˆ T k(z)). Assume Q′ ⊂ ˆ ∆ℓ. The points x0 = ˆ T −ℓ( ˆ T k(x)) and z0 = ˆ T −ℓ( ˆ T k(z)) are in a same unstable disk of ˆ ∆0. Set t = s(f R(x0), f R(z0)). We have t = s(x0, z0) − 1 = s( ˆ T k(x), ˆ T k(z)) − 1 ≥ bk( ˆ T k(x)) − 1 Using (14) and (Y3), we may write dist(π( ˆ T k(x)), π( ˆ T k(z))) = dist(π( ˆ T ℓ(x0)), π( ˆ T ℓ(z0))) = dist(f ℓ(x0), f ℓ(z0)) ≤ C dist(f R(x0), f R(z0)) ≤ Cβt dist((f R)t(x0), (f R)t(z0)) ≤ Cβbk( ˆ

T k(x))−1 dist((f R)t(x0), (f R)t(z0)).

Altogether with (25), (26) and (29) it yields Lemma 3.11 with σ = βη.

77

Define for each k ≥ 1 Rk =

k−1

• j=0

R ◦ F j. (30)

Proposition 3.12

Given 0 < σ < 1, there exists C > 0 such that for all k ≥ 1 we have

• σbkdν ≤ C
• ℓ≥k/3

mγ0{R ≥ ℓ} + Ck

• ℓ≥1

σℓmγ0

• Rℓ > k

3

• .

It is enough to have the result with ν0 playing the role of mγ0, where ν0 is the F-invariant measure ≪ mγ0. We have for each n ≥ 1

• σbndν =

∞

• k=0

σkν{bn = k}. (31)

Exercise

ν|∆0 = ν(∆0)ν0 (32)

78

For a point x with bn(x) ≥ 1, we define rn(x) = min{1 ≤ j ≤ n : T j(x) ∈ ∆0} and sn(x) = max{1 ≤ j ≤ n : T j(x) ∈ ∆0}. For each k ≥ 1 we may write {bn = k} ⊂

• 1≤i≤j≤n

{bn = k, rn = i, sn = j} =

• 1≤i<n/3

2n/3<j≤n

{bn = k, rn = i, sn = j} ∪

• rn ≥ n

3

• ∪
• sn ≤ 2n

3

• .

(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{rn = i} = ν{R ≥ i} = ν(∆0)ν0{R ≥ i}. Hence ν

• rn ≥ n

3

• ≤ ν(∆0)
• i≥n/3

ν0{R ≥ i}. (34) On the other hand, if j = sn(x) ≤ 2n/3, then T n−j(x) ∈ ∆n−j and n − j ≥ n/3. Using once more the T-invariance of ν and (32) ν

• sn ≤ 2n

3

• ≤ ν
• T −n+j (∆n−j)
• = ν (∆n−j) ≤ ν(∆0)
• i≥n/3

ν0{R ≥ i}. (35)

80

Finally,

• 1≤i<n/3

2n/3<j≤n

ν {bn = k, rn = i, sn = j} =

• 1≤i<n/3

2n/3<j≤n

ν

• bj−i ◦ T i = k − 1, rn = i, sn = j
• ≤
• 1≤i<n/3

2n/3<j≤n

ν

• T −i {bj−i = k − 1} ∩ T −i(∆0) ∩ T −j(∆0)
• =
• 1≤i<n/3

2n/3<j≤n

ν

• {bj−i = k − 1} ∩ ∆0 ∩ T −j+i(∆0)
• ≤ n
• ℓ>n/3

ν

• {bℓ = k − 1} ∩ ∆0 ∩ T −ℓ(∆0)
• = nν(∆0)
• ℓ>n/3

ν0

• {Rk = ℓ} ∩ ∆0 ∩ T −ℓ(∆0)
• ≤ nν(∆0)ν0
• Rk > n

3

• .

(36)

81

Fom (33), (34), (35) and (36), we obtain for k ≥ 1 ν{bn = k} ≤ ν(∆0)  2

• i>n/3

ν0{R ≥ i} + nν0

• Rk ≥ n

3

• 

 . (37) Now, observe that bn(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{bn = 0} =

• j≥n

ν{R > j} = ν(∆0)

• j≥n

ν0{R > j} ≤ ν(∆0)

• j>n/3

ν0{R ≥ j}.

Together with (31) and (37) this gives the conclusion.

Corollary 3.13

For every H¨

• lder continuous ϕ : M → R and k ≥ 1

ˆ ϕ ◦ ˆ T k − ϕk1 ≤ C

• ℓ≥k/3

mγ0{R ≥ k} + Ck

• ℓ≥1

σℓmγ0

• Rℓ > 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 TAM = 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 lim inf

n→+∞

1 n

n

• j=1

log Df −1|E cu

f j (x) < −c.

(NUE)

Theorem 4.1 (Alves, Dias, Luzzatto, and Pinheiro 2017)

Assume A is a partially hyperbolic attractor with TAM = 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 µ1, ..., µℓ on A such that m almost every x ∈ H belongs in the basin of µj, 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

• j=n−k+1

Df −1 | E cu

f j(x) ≤ σk,

for all 1 ≤ k ≤ n. As a consequence of Pliss Lemma, we have:

Lemma 4.2

There exist σ > 0 and 0 < θ ≤ 1 such that if 1 n

n−1

• i=0

log Df −1 | E cu

f i(x) < −c

then x has at least θn hyperbolic times. Fix σ as in the lemma and define Hn =

• x ∈ H : n is a hyperbolic time for x
• .

(38)

85

cu-disks

The dominated splitting TAM = 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 TxD ⊂ C cu

x

for all x ∈ D. 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 mD(H) > 0. We fix once and for all a cu-disk D in these conditions.

86

Lemma 4.3

There exist C, δ1 > 0 such that for each x ∈ Hn ∩ D there exists a neighborhood Vn(x) of x in D so that:

1 f n maps Vn(x) to a disk of radius δ1 centred at f n(x); 2 for all 1 ≤ k ≤ n and y, z ∈ Vn(x),

distf n−k(Vn(x))(f n−k(y), f n−k(z)) ≤ σk/2 distf n(Vn(x))(f n(y), f n(z));

3 for all y, z ∈ Vn(x)

log | det Df n|TyD| | det Df n|TzD| ≤ C distf n(D)(f n(y), f n(z)). The sets Vn(x) will be called hyperbolic pre-disks.

87

Local unstable manifolds

Lemma 4.4

There are hyperbolic pre-disks V1, V2, · · · ⊂ D and integers n1 < n2 < · · · such that for Bk = f nk(Vk) lim

k→∞

mBkf nk(H ∩ D) mBk(Bk) = 1. Considering an accumulation point of the sequence of the centers of the cu-disks (Bk)k, by Ascoli-Arzel` a, there is an accumulation disk ∆ of (Bk)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 Ω1, ..., Ωℓ ⊂ A such that for Lebesgue almost every x ∈ H we have ω(x) = Ωj for some 1 ≤ j ≤ ℓ. Moreover, each Ωj 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 := Ωj and ∆ := ∆j ⊂ Ωj, with Ωj and ∆j as in the theorem. Set m0 = m∆.

89

Cylinders

We fix a constant δs > 0 so that stable disks W s

δs(x)

are defined for all points x ∈ Ω. Given a subdisk ∆′ ⊂ ∆, define C(∆′) =

• x∈∆′

W s

δs(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 Vn(x) ⊂ ∆, there is 0 ≤ ℓ ≤ L such that f n+ℓ(Vn(x)) u-crosses C(Bu

δ0(p)).

Consider ∆0 = Bu

δ0(p).

For each x ∈ Hn there are 0 ≤ ℓ ≤ L and a cu-disk ωx

n ⊂ Vn(x) s.t.

π(f n+ℓ(ωx

n)) = ∆0.

(∗)

91

Inducing scheme

We inductively define an m0 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 n0 ∈ N and choose a maximal set of points x1, . . . , xjn0 ∈ ∆0 ∩ Hn0 such that the sets ωxi

n0

are pairwise disjoint and contained in ∆0. Let Pn0 = {ωx1

n0, . . . , ω xjn0 n0 }.

These are the elements of the partition P constructed in the first step. For each 0 ≤ i ≤ jn0, we define the return time R|ωxi

n0 = n0 + ℓi,

where 0 ≤ ℓi ≤ L is the integer associated to ωxi

n0 as in (∗). 92

Let Zn0 be the set of points z ∈ ∆0 ∩ Hn0 such that ωz

n0 could have been

(but were not) chosen in the construction of Pn0. For every z ∈ Zn0, the set ωz

n0 must either intersect some ωxi n0 ∈ Pn0 or

∆c

0 := ∆ \ ∆0 (for otherwise it would have been included in Pn0 ). 93

Satellites

For each ω ∈ Pn0 ∪ {∆c

0} we define

Z ω

n0 =

• x ∈ Zn0 : ωx

n0 ∩ ω = ∅

• and the n0-satellite associated to ω

Sω

n0 =

• x∈Z ω

n0

Vn0(x). Finally we consider Sn0 =

• ω∈Pn0∪{∆c

0}

Sω

n0.

By construction we have Hn0 ⊂ Sn0 ∪

• ω∈Pn0∪{∆c

0}

ω.

94

Inductive step

Assume now that the construction has been carried out up to time n − 1 for some n > n0: we have a collection of pairwise disjoint sets Pk = {ωx1

k , ..., ω xjk k },

for each n0 ≤ k ≤ n − 1, such that for any k = k′ ω ∈ Pk and ω′ ∈ Pk′ ⇒ ω ∩ ω′ = ∅. Define ∆n−1 = ∆0 \

• ω∈Pn0∪···∪Pn−1

ω.

95

We choose a maximal subset

• f points x1, . . . , xkn in Hn such

that the corresponding ωxi

n are pairwise

disjoint and contained in ∆n−1. Define Pn = {ωx1

n , . . . , ωxjn n }

For each 0 ≤ i ≤ jn, let 0 ≤ ℓi ≤ L be as in (∗) and set R|ω

xi n = n + ℓi.

Let Zn be the set of points z ∈ ∆n−1 ∩ Hn such that ωz

n could have been

chosen in the construction of Pn. Given ω ∈ Pn0 ∪ · · · ∪ Pn ∪ {∆c

0}, define

Z ω

n = {z ∈ Zn : ωz n ∩ ω = ∅}

and its n-satellite Sω

n =

• x∈Z ω

n

Vn(x). Finally, define Sn =

• ω∈Pn0∪···∪Pn∪{∆c

0}

Sω

n . 96

By construction, we have for each n ≥ n0 Hn ⊂ Sn ∪

• ω∈Pn0∪···∪Pn∪{∆c

0}

ω. (∗∗) Define ∆n = ∆0 \

• ω∈Pn0∪···∪Pn

ω. and P =

• n≥n0

Pn.

97

Estimates

Lemma 4.7

There is C > 0 such that for all k ≥ n0, ω ∈ Pk ∪ {∆c

0} and n ≥ k

m0(Sω

n ) < Cσ

n−k 2 m0(ω).

Lemma 4.8

∞

• n=n0

m0(Sn) < ∞. Thus, by Borel-Cantelli Lemma, m0 almost every x ∈ ∆0 belongs in finitely many Sn’s. Since m0 almost every x ∈ ∆0 has infinitely many hyperbolic times and (∗∗) holds, then m0 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 : γ0 ∩ Λ → γ0 ∩ Λ its quotient map. Set ∆0 = γ0 ∩ Λ and m0 = mγ0. Let (Hn)n be as in (38). Note that there exists θ > 0 such that for m0 almost every x ∈ ∆0 lim sup

n→∞

1 n# {1 ≤ j ≤ n : x ∈ Hj} > θ. (39) Set R0 = 0 and Rk = k−1

j=0 R ◦ F j, for each k ≥ 1. We have

x ∈ Hn = ⇒ F i(x) ∈ Hn−Ri, (40) whenever Ri(x) ≤ n < Ri+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 (Hn)n of sets in ∆0 for which (39) + (40) hold; 2 a sequence (Sn)n of sets in ∆0 such that

n≥1 m0(Sn) < ∞;

3 L ∈ N such that Hn ∩ {R > L + n} ⊂ Sn, for all n ≥ 1.

Then, R is integrable with respect to m0.

99

Consider the ergodic F-invariant probability measure ν ≪ m0. It is enough to check the integrability of R with respect to ν. Assume by contradiction that R / ∈ L1(ν). Since R is positive, Birkhoff’s Ergodic Theorem gives lim

k→∞

1 k

k−1

• i=0

R(F i(x)) →

• Rdν = ∞,

(41) for ν almost every x ∈ ∆0. Since

n≥1 m0(Sn) < ∞, it follows from

Borel-Cantelli Lemma that ν almost every x ∈ ∆0 belongs in a finite number of sets Sn. Define s(x) = # {n ≥ 1 : x ∈ Sn} for x ∈ ∆0. We have for ν almost every x ∈ ∆0 1 k

k−1

• i=0

s(F i(x)) →

• sdν =
• n≥1

ν(Sn) < ∞. (42)

100

Given i ≥ 0 and Ri ≤ j < Ri+1, we have F i(x) ∈ Hj−Ri, whenever x ∈ Hj. We have R(F i(x)) ≥ j − Ri, for otherwise we would have Ri+1 − Ri = R(F i(x)) < j − Ri ≤ Ri+1 − Ri. Set k = j − Ri. Since Hk ∩ {R > k + L} ⊂ Sk, we have F i(x) ∈ Sk or R(F i(x)) = k + ℓ for some 0 ≤ ℓ ≤ L. So, the number of integers j with Ri ≤ j < Ri+1 for which x ∈ Hj is bounded by the number of integers k such that F i(x) ∈ Sk or F i(x) ∈ {R = k + ℓ}, for some 0 ≤ ℓ ≤ L. Thus # {Ri ≤ j < Ri+1 : x ∈ Hj} ≤ 1 + s(F i(x)). Given n ≥ 1, define r(n) = min{Ri : Ri > n}. For each n ≥ 1, we have #{1 ≤ j ≤ n : x ∈ Hj} ≤

r(n)

• i=0

(1 + s(F i(x))) ≤ r(n) +

r(n)

• i=0

s(F i(x)). Therefore, 1 n# {j ≤ n : x ∈ Hj} ≤ r(n) n  1 + 1 r(n)

r(n)

• i=0

s(F i(x))   . (43)

101

Observe that if r(n) = k, then by definition we have Rk−1 ≤ n < Rk. Hence, Rk−1 k ≤ n r(n) < Rk k = Rk k + 1

• 1 + 1

k

• ,

which together with (41) gives lim

n→∞

n r(n) = lim

k→∞

Rk k = lim

k→∞

1 k

k−1

• i=0

R(F i(x)) = ∞. (44) It follows from (42), (43) and (44) that lim

n→∞

1 n# {1 ≤ j ≤ n : x ∈ Hj} = lim

n→∞

r(n) n = 0, which clearly contradicts (39).

102

Decay of correlations

Recall that f is strongly nonuniformly expanding along E cu on a set H ⊂ U if there is c > 0 and a Riemannian metric on M such that for all x ∈ H lim sup

n→+∞

1 n

n

• j=1

log Df −1|E cu

f j (x) < −c.

(SNUE) If SNUE holds for x, then we have defined E(x) = min

• N ≥ 1: 1

n

n−1

• i=0

log Df −1 | E cu

f i(x) < −c,

∀ n ≥ N

• Theorem (Alves and Pinheiro 2010; Alves and Li 2015)

1 If there is an unstable disk γ ⊂ A with mγ{E > n} n−α for some

α > 0, then f N has an SRB measure µ for some N ≥ 1, and Corµ(ϕ, ψ ◦ f Nn) n−α+1 for ϕ, ψ ∈ H.

2 If there is an unstable disk γ ⊂ A such that mγ{E > n} e−cnθ for

some c > 0 and 0 < θ ≤ 1, then f N has an SRB measure µ for some N ≥ 1, and Corµ(ϕ, ψ ◦ f Nn) e−c′nθ for ϕ, ψ ∈ H.

103

Tail estimates

Observe that as there is C > 0 such that S∆c

n

⊂ {x ∈ ∆0 | distD(x, ∂∆0) ≤ Cσn/2} then mD

• S∆c

n

• σn/2.

(45) Let 0 < θ < 1 be given by Pliss Lemma and x ∈ ∆n such that x / ∈ {E > n} ∪

• x | distD(x, ∂∆0) ≤ Cσ

θn 2

• .

Recall that for n large, x has at least θn hyperbolic times between 1 and n, then at least θn

2 between θn 2 and n. Denote them by t1 < · · · < tk ≤ n.

As distD(x, ∂∆0) ≥ Cσ

θn 2 , we have x ∈ Sti(∆0) for i = 1, . . . , k. Thus,

x ∈ X θn 2 , n

• :=
• x ∈ ∆n | ∃t1 < · · · < t θn

2 ≤ n, x ∈ θn 2

• i=1

Sti(∆0)

• .

104

Hence, we have ∆n ⊂ {E > n} ∪

• x ∈ ∆0 | distD(x, ∂∆0) ≤ Cσ

θn 2

• ∪ X

θn 2 , n

• .

Set for integers k, n X(k, n) =

• x ∈ ∆n | ∃t1 < · · · < tk ≤ n, x ∈

k

• i=1

Sti(∆0)

• .

Lemma (Gou¨ ezel 2006; Alves and Li 2015)

There exist C > 0 and λ < 1 such that for all n and 1 ≤ k ≤ n, mD(X(k, n)) ≤ CλkmD(∆0).

105

Thank You!

106

References I

Alves, J. F., C. Bonatti, and M. Viana (2000). “SRB measures for partially hyperbolic systems whose central direction is mostly expanding”.

• Invent. Math. 140.2, pp. 351–398.

Alves, J. F., C. L. Dias, S. Luzzatto, and V. Pinheiro (2017). “SRB measures for partially hyperbolic systems whose central direction is weakly expanding”. J. Eur. Math. Soc. (JEMS) 19.10, pp. 2911–2946. Alves, J. F. and X. Li (2015). “Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors”. Adv.

• Math. 279.0, pp. 405 –437.

Alves, J. F., S. Luzzatto, and V. Pinheiro (2005). “Markov structures and decay of correlations for non-uniformly expanding dynamical systems”.

• Ann. Inst. H. Poincar´

e Anal. Non Lin´ eaire 22.6, pp. 817–839. Alves, J. F. and V. Pinheiro (2008). “Slow rates of mixing for dynamical systems with hyperbolic structures”. J. Stat. Phys. 131.3, pp. 505–534.

107

References II

Alves, J. F. and V. Pinheiro (2010). “Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction”. Adv. Math. 223.5, pp. 1706–1730. Bonatti, C. and M. Viana (2000). “SRB measures for partially hyperbolic systems whose central direction is mostly contracting”. Israel J. Math. 115, pp. 157–193. Bowen, R. (1975). Equilibrium states and the ergodic theory of Anosov

• diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin:

Springer-Verlag, pp. i+108. Bowen, R. and D. Ruelle (1975). “The ergodic theory of Axiom A flows”.

• Invent. Math. 29.3, pp. 181–202.

Buzzi, J. and O. Sarig (2003). “Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps”. Ergodic Theory Dynam. Systems 23.5, pp. 1383–1400.

108

References III

Castro, A. (2002). “Backward inducing and exponential decay of correlations for partially hyperbolic attractors”. Israel J. Math. 130,

• pp. 29–75.

Castro, A. (2004). “Fast mixing for attractors with a mostly contracting central direction”. Ergodic Theory Dynam. Systems 24.1, pp. 17–44. Dolgopyat, D. (2000). “On dynamics of mostly contracting diffeomorphisms”. Comm. Math. Phys. 213.1, pp. 181–201. Gou¨ ezel, S. (2005). “Berry-Esseen theorem and local limit theorem for non uniformly expanding maps”. Ann. Inst. H. Poincar´ e Probab. Statist. 41.6, pp. 997–1024. Gou¨ ezel, S. (2006). “Decay of correlations for nonuniformly expanding systems”. Bull. Soc. Math. France 134.1, pp. 1–31. Hu, H. (2004). “Decay of correlations for piecewise smooth maps with indifferent fixed points”. Ergodic Theory Dynam. Systems 24.2,

• pp. 495–524.

109

References IV

Korepanov, A., Z. Kosloff, and I. Melbourne (2019). “Explicit coupling argument for non-uniformly hyperbolic transformations”. Proc. Roy.

• Soc. Edinburgh Sect. A 149.1, pp. 101–130.

Krzy˙ zewski, K. and W. Szlenk (1969). “On invariant measures for expanding differentiable mappings”. Studia Math. 33, pp. 83–92. Ledrappier, F., Y. Lima, and O. Sarig (2016). “Ergodic properties of equilibrium measures for smooth three dimensional flows”. Comment.

• Math. Helv. 91.1, pp. 65–106.

Liverani, C., B. Saussol, and S. Vaienti (1999). “A probabilistic approach to intermittency”. Ergodic Theory Dynam. Systems 19.3, pp. 671–685. Melbourne, I. (2009). “Large and moderate deviations for slowly mixing dynamical systems”. Proc. Amer. Math. Soc. 137.5, pp. 1735–1741. Melbourne, I. and M. Nicol (2005). “Almost sure invariance principle for nonuniformly hyperbolic systems”. Comm. Math. Phys. 260.1,

• pp. 131–146.

110

References V

Melbourne, I. and M. Nicol (2008). “Large deviations for nonuniformly hyperbolic systems”. Trans. Amer. Math. Soc. 360.12, pp. 6661–6676. Melbourne, I. and M. Nicol (2009). “A vector-valued almost sure invariance principle for hyperbolic dynamical systems”. Ann. Probab. 37.2, pp. 478–505. Oseledec, V. I. (1968). “A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems”. Trudy Moskov. Mat. Obˇ sˇ

• c. 19, pp. 179–210.

Pesin, J. B. (1976). “Families of invariant manifolds that correspond to nonzero characteristic exponents”. Izv. Akad. Nauk SSSR Ser. Mat. 40.6, pp. 1332–1379, 1440. Pesin, Y. B. and Y. G. Sinai (1982). “Gibbs measures for partially hyperbolic attractors”. Ergodic Theory Dynam. Systems 2.3-4, 417–438 (1983). Pinheiro, V. (2006). “Sinai-Ruelle-Bowen measures for weakly expanding maps”. Nonlinearity 19.5, pp. 1185–1200.

111

References VI

Pliss, V. A. (1972). “On a conjecture of Smale”. Differencial’nye Uravnenija 8, pp. 268–282. Ruelle, D. (1976). “A measure associated with axiom-A attractors”.

• Amer. J. Math. 98.3, pp. 619–654.

Sarig, O. (2003). “Existence of Gibbs measures for countable Markov shifts”. Proc. Amer. Math. Soc. 131.6, pp. 1751–1758. Sarig, O. M. (1999). “Thermodynamic formalism for countable Markov shifts”. Ergodic Theory Dynam. Systems 19.6, pp. 1565–1593. Sarig, O. M. (2013). “Symbolic dynamics for surface diffeomorphisms with positive entropy”. J. Amer. Math. Soc. 26.2, pp. 341–426. Sinai, J. G. (1972). “Gibbs measures in ergodic theory”. Uspehi Mat. Nauk 27.4, pp. 21–64. Young, L.-S. (1998). “Statistical properties of dynamical systems with some hyperbolicity”. Ann. of Math. (2) 147.3, pp. 585–650. Young, L.-S. (1999). “Recurrence times and rates of mixing”. Israel J.

• Math. 110, pp. 153–188.

112