Some statistical properties of almost Anosov diffeomorphisms

Chaos, Solitons and Fractals 123 (2019) 149–162

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Some statistical properties of almost Anosov diffeomorphisms Xu Zhang Department of Mathematics, Shandong University, Weihai, Shandong 264209, China

a r t i c l e

i n f o

Article history: Received 19 October 2018 Revised 25 March 2019 Accepted 2 April 2019

Keywords: Almost Anosov diffeomorphism Correlation function Decay rate Sinai-Ruelle-Bowen measure

a b s t r a c t For a kind of almost Anosov diffeomorphisms, we study the relationship among the existence of SinaiRuelle-Bowen (SRB) measures, the local differentiability near the indifferent fixed points, and space dimension, where the almost Anosov diffeomorphisms are hyperbolic everywhere except for the indifferent fixed points. As a consequence, there are C2 almost Anosov diffeomorphisms that admit σ -finite (infinite) SRB measures in spaces with dimensions bigger than one; there exist C2 almost Anosov diffeomorphisms with finite SRB measures in spaces with dimensions bigger than three. Further, we obtain the lower and upper polynomial bounds for the decay rates of the correlation functions of the Hölder observables for the maps admitting finite SRB measures.

1. Introduction The existence of the invariant measures and the minimal requirements of the differentiability of the maps are two basic problems in smooth ergodic theory [33]. One important result is that a C2 Anosov diffeomorphism on a compact connected Riemannian manifold has an invariant measure, which has absolutely continuous conditional measures on unstable manifolds [46]. This was generalized to Axiom-A systems [15]. Based on these work, a kind of invariant measures with absolutely continuous conditional measures on unstable manifolds are called Sinai-Ruelle-Bowen (SRB) measures, which are closely related to chaotic attractors, and have some good dynamical properties in physics [17,51]. Lots of results on the existence of SRB measures have been obtained, for example, the non-uniformly hyperbolic maps [10], the singular maps [28], the billiard systems (the maps defined as the reflection on the boundary of the billiard) [11,19,29], the continuous Lorenz system (a Poincaré map exhibiting an invariant foliation) [48], the geometric Lorenz attractor proved by Pesin (unpublished), and the singular hyperbolic flows [6]. On the other hand, there exist many maps, which do not admit finite SRB measures, for instance, a C2 piecewise expanding one-dimensional map T: [0, 1] → [0, 1] with the derivative at one fixed point equal to one [39], a C2 almost Anosov diffeomorphisms defined on a twodimensional manifold [27]. Since these maps have infinite invariant measures, the σ -finite SRB measures were introduced, which are invariant and have absolutely continuous conditional measures on unstable manifolds (See [25, Definition 4] or Definition 2.4).

E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2019.04.005 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

The correlation function for a dynamical system provides information on the mixing speed, this function decays to zero as the evolution of the system, assuming the system is mixing. In physics, an observable is a dynamical variable that can be measured. The correlation function illustrates the independence of two observables on the evolution of the systems. There exist many results on the estimates of the bounds for the decay rates of the correlation functions. A systematic method was introduced, namely “Young tower”, to obtain the exponential bounds for some systems, like logistic map, piecewise hyperbolic systems, and scattering billiards [49]. The coupling method was introduced to get the upper polynomial bounds [50], whereas the lower polynomial bounds were obtained by the renewal theory [21,36,44]. And, there are many interesting results on some particular maps, for instance, the exponential decay rates for Hénon map [12], the upper polynomial decay rates for a class of area preserving maps [32], the upper polynomial decay rates for some hyperbolic systems with one center unstable direction [22]. On the other hand, for the study of decay rates for the correlation functions of flows, please refer to [5,8,9,34,35,37]. Two natural questions are related to an SRB measure: “What are the main factors or parameters that determine the existence of an SRB measure?”, “What are the decay rates of the correlation functions with respect to an SRB measure?”, where an SRB measure is a natural candidate for the study of the correlation functions, since it is invariant and has good physical meaning. For some almost Anosov diffeomorphisms, we study these two problems. We discuss the relationship among the differentiability, the dimension of the manifold, and the existence of a finite/σ -finite SRB measure, and obtain the lower and upper polynomial bounds for the decay rates of the correlation functions

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X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

of Hölder observables for maps admitting finite SRB measures. As a consequence, there are C2 almost Anosov diffeomorphisms that admit σ -finite SRB measures in spaces with dimensions bigger than one; there exist C2 almost Anosov diffeomorphisms that admit SRB measures in spaces with dimensions bigger than three. As far as we know, our work is the first one that investigates the relationship between the dimensions and the existence of SRB measures. We show the existence of diffeomorphisms that admit σ -finite SRB measures in spaces with dimensions bigger than one, so our results could be thought of as the generalization of the work in [27]. The difficult and important part of our construction of the SRB measures is the distortion estimate (Proposition 3.3), which plays an important role in the proof that the invariant measures have absolutely continuous conditional measures on unstable manifolds. We combine the idea in [27] and some elegant analysis to solve this problem. The rest is organized as follows. In Section 2, some concepts and the main results are introduced. In Section 3, the existence of SRB or σ -finite SRB measures in spaces with dimensions bigger than or equal to two is studied. This section is divided into three parts. In Section 3.1, the properties of the stable and unstable manifolds are studied. In Section 3.2, the distortion estimates are considered. In Section 3.3, the finite or σ -finite SRB measures are constructed. In Section 4, the lower and upper polynomial bounds are obtained. This section consists of three parts. In Section 4.1, a quotient map by collapsing the stable manifolds in each element of the Markov partition is defined. In Section 4.2, both the lower and upper polynomial bounds for the decay rates of the correlation functions for the quotient system are obtained, where the observables are defined on the quotient manifolds. In Section 4.3, the polynomial bounds for Hölder observables for the original diffeomorphisms are obtained. 2. Main results In this section, some concepts and the main results are introduced. We introduce a notation to represent some kinds of norms | · |object , where the subscript is related to the object inside the two vertical line segments, for example, if x is an element in an Euclidean space, then |x|e represents the Euclidean norm; if x is an operator, then |x|o is the norm of the operator; for convenience, if x is a real number, then |x| is the absolute value. For an operator T and a set V, |T |V |o = supv∈V ||Tvv||∗ , where | · |∗ is the norm defined ∗ on V. Definition 2.1 [20,42]. Given a measurable space X with a probability measure ν and a measurable partition ξ , consider a map : x ∈ X → ξ defined by (x ) = ξ (x ), where ξ (x) is an element of the partition containing the point x. Set μ := ∗ ν = ν ◦ −1 . By the measure disintegration theorem, there exists a Borel map y → ν y such that for μ almost every y ∈ ξ , νy (−1 (y )) = 1 and  ν = ξ νy dμ(y ), that is, for every Borel subset B of X,

ν (B ) =



ξ

νy ( B ) d μ ( y ) .

This family {ν y , y ∈ ξ }, denoted by ν ξ for convenience, is said to be a canonical system of conditional measures for ν and ξ . Let M be a C∞ compact Riemannian manifold without boundary. Let ν be the Riemannian measure on M. For a map f on M, if |D fxn |o grows exponentially fast with n, then we call the map f has a positive Lyapunov exponent at x ∈ M, where D fxn is the derivative operator of the nth iteration of f at the point x. For more information on Lyapunov exponents, please refer to [33]. For a C 1+α∗ (α ∗ > 0) measurable map f on M, assume that μ is an invariant Borel probability measure, and f has positive

Lyapunov exponents for μ-almost every x ∈ M. It follows from the Pesin theory [38] that the unstable manifold Wu (x) exists for μ-almost every x ∈ M, and is an immersed submanifold of M. Denote by νxu the Riemannian measure induced on Wu (x). Given a measurable partition ξ , if ξ (x) ⊂ Wu (x) and ξ (x) contains an open neighborhood of x in Wu (x) for μ-almost every x ∈ M, then ξ is said to be subordinate to the unstable manifolds; further, if μξ is absolutely continuous with respect to νxu for μ-almost every x ∈ M, then we call the measure μ has absolutely continuous conditional measures on the unstable manifolds, where μξ is a canonical system of conditional measures for μ and ξ [30]. Definition 2.2. An invariant Borel probability measure μ for the map f on M is said to be an SRB measure if (a) f has positive Lyapunov exponents almost everywhere with respect to the measure μ; (b) μ has absolutely continuous conditional measures on unstable manifolds with respect to the Riemannian measure induced on unstable manifolds. Definition 2.3. Given a subset V ⊂ M, the first return map to MV is defined by g = f τ (x ) (x ) : M \ V → M \ V, where τ (x ) = min{i > 0 : f i (x ) ∈ M \ V } is the first return time function with respect to the set V. Definition 2.4 [25, Definition 4]. An infinite invariant Borel measure μ for the map f on M is said to be a σ -finite SRB measure if (i) there is a set E so that for any open neighborhood V of the set E, one has 0 < μ(MV) < ∞; (ii) the first return map defined on the set MV has positive Lyapunov exponents almost everywhere with respect to the measure μ; (iii) the conditional measures of μ on unstable manifolds are absolutely continuous with respect to the Riemannian measure induced on unstable manifolds. Remark 2.1. Here, we show that the infinite measure in [27] is a σ -finite SRB measure, since the definition of the σ -finite SRB measures was not used in [27]. For convenience, we use the same notations as in [27]. Consider the diffeomorphism f with a fixed point p in [27]. Suppose P is a small rectangle containing p (for the definition of rectangle, please see Definition 3.1), the first return map is defined by g = f τ with respect to MP. Let E = { p} and V = P . The construction of the σ -finite SRB measure is given in the proof of [27, Theorem C]. Properties (i) and (iii) of Definition 2.4 are contained in the conclusion part of [27, Theorem C]. Property (ii) follows from [27, Lemmas 5.2 and 5.3] and the arguments for the construction of the invariant measure in the proof of [27, Theorem C]. Definition 2.5 [43]. Given a measurable space (X, B ) and a signed measure μ, the upper variation and lower variation for any measurable set B ∈ B are defined as

W (μ, B ) = sup{μ(E ) : E ∈ B and E ⊂ B} and

W (μ, B ) = inf{μ(E ) : E ∈ B and E ⊂ B}, respectively. The variation of the signed measure μ is a function

|μ|m (B ) = W (μ, B ) + |W (μ, B )|, ∀B ∈ B. The total variation is the value of the variation on the whole space, that is, |μ|m = |μ|m (X ). Remark 2.2. Definition 2.5 is used in Section 4.3. Now, we introduce the almost Anosov maps considered here. The map has only one indifferent fixed point, and the Jacobian at

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

the indifferent fixed point has exactly m − 1 eigenvalues equal to one, and 1 eigenvalue with the absolute value less than one, where m is the dimension of the space. Assume M is a C∞ compact Riemannian manifold without boundary, ν is the Riemannian measure on M, and the dimension of M is m ≥ 2. Let f be a diffeomorphism defined on M satisfying the following properties: (1) Assume that f has a fixed point p, f is topologically mixing and topologically conjugate to an Anosov diffeomorphism; (2) there exist a constant 0 < κ s < 1 and a continuous function κ u with



=1 >1

κ u (x )

at x = p elsewhere,

and there is a decomposition of the tangent space Tx M:

Tx M = Exu  Exs ,

v∈Exs ,v =0

|D f x v|r ≤ κ s, |v|r

(2(ii)) κ u (x) is the infimum of the expanding rate

inf

v∈Exu ,v =0

|D f x v|r ≥ κ u ( x ), |v|r

(2(iii)) the supremum of the expanding rate is bounded by 1/κ s

sup x∈M,v

∈Exu ,

1 |D f x v|r ≤ s; | v | κ r v =0

(3) dimExu = m − 1 and dimExs = 1; (4) there is a small neighborhood U of p, on which f is written as

f (x1 , x2 , . . . , xm−1 , xm ) = (h(x1 , . . . , xm )x1 , . . . , h(x1 , ..., xm )xm−1 , κs xm ), where



h ( x1 , . . . , xm ) = 1 +

 {2} ∪ ( 3, +∞ ), α∈ ( 0, +∞ ),

m −1 

(2.1)

 x2i α /2 + ρ x2m ,

ρ = 0,

i=1

if m ≥ 3 if m = 2;

(2.2)



C 1+α , C2 ,

Remark 2.6. Note that the map f is global C2 except for the case m = 2 and 0 < α ≤ 1. The requirement (2.2) guarantees that f is at least C2 (See Proposition 3.7). The conclusion that local expression (2.1) is C 1+α for 0 < α ≤ 1 could be derived by an inequality |xα − yα | ≤ |x − y|α for x, y ∈ [0, ∞) and 0 < α ≤ 1. By (2.1), if the value of α is increasing, then the local differentiability of the map near the fixed point p will be improved. When α varies, the local dynamics do not change in the sense of topological conjugacy, however the measures of the subsets of MU with the same first return time to MU change, this fact affects the existence of SRB measures (This is the key idea in the construction of the SRB measures. For details, see the proof of Theorem 2.1). Remark 2.7. The assumption dimE s = 1 plays an important role in the proof of Proposition 3.2, which is critical in the study of the SRB measures. The term ρ x2m in (2.1) can be replaced by other differentiable function ψ (xm ) satisfying ψ (0 ) = 0 and ψ (0 ) = 0. Remark 2.8. Some criteria on the existence of SRB measures, e.g. the results in [2–4], could not be applied in our present work, since it is difficult to check the limits of the sequence 1 n −1 | cu | , where x is taken from a set of posio E j=1 log |D f n f j (x )

(5) if m = 2,

f is

Jacobian at this fixed point very close to one, we perturb this fixed point to an indifferent fixed point such that the perturbed map is an almost Anosov map, and the perturbed almost Anosov map is topologically conjugate with the original Anosov map. Locally, one could think of the fixed point is (0,0), the Anosov map is written as (λx, ηy) with |λ| > 1 and |η| < 1, where (x, y) is in a small neighborhood of (0,0). Next, one only needs to perturb λ and η to 1 smoothly. The topological conjugacy is obviously kept. In particular, the construction of almost Anosov diffeomorphisms can be obtained by isotopy from Anosov diffeomorphisms in dimension two. In the higher dimensional context, for instance in dimension three, it could be that the initial Anosov diffeomorphisms present only complex eigenvalues along Eu , and it follows from the work of [14] (in the C1 -topology) or [13] (in the Cr -topology) that one could construct almost Anosov diffeomorphisms satisfying (2.1). Remark 2.5. The parameter α has a similar role as the parameter α 0 in the one-dimensional Pomeau-Manneville map g(x ) ≈ x + x1+α0 [39]. However, the map introduced here has another parameter, “dimension”, that affects the existence of finite or σ -finite SRB measures.

such that (2(i)) the contracting rate is bounded by κ s

|D fx |Exs | = sup

151

if 0 < α ≤ 1 if α > 1;

if m ≥ 3, f is C2 . Remark 2.3. Note that M is a Riemannian manifold, in Assumption (2), for any v in the tangent space, |v|r is the norm given by the Riemannian metric on the tangent space, |D fx |Exs |o is the norm of the operator Dfx restricted to Exs . Remark 2.4. The parameters m and α are related to the existence of finite or σ -finite SRB measures. There are many examples satisfying the above assumptions. Consider an Anosov map X = AX on Tm , where A is an m × m integer matrix with determinant ± 1, the smallest absolute value of the eigenvalues of A is less than one, and the absolute values of the other m − 1 eigenvalues are bigger than 1. By properly choosing A and small perturbations, we can obtain almost Anosov maps satisfying the above assumptions. For example, suppose there is an Anosov map on a two dimensional torus, take a fixed point with the norm of the eigenvalues of the

tive Lebesgure measure on which f is non-uniformly expanding along Ecu , and Ecu is the combination of the central and unstable direction. Note that the assumption in [3] is weaker than  that in [2], where lim infn→∞ 1n nj=1 log |D f −1 |E cu |o < − and f j (x )  lim supn→∞ 1n nj=1 log |D f −1 |E cu |o < − are used in [2,3], respecf j (x )

tively, where > 0 is a constant.

Theorem 2.1. Let f be a diffeomorphism satisfying the above assumptions. If α < m − 1, then there exists an SRB measure; if α ≥ m − 1, then there is a σ -finite SRB measure. By Theorem 2.1, we have the following corollary. Corollary 2.1. There are C2 almost Anosov diffeomorphisms with SRB measures in spaces with dimensions bigger than three; there exist C2 almost Anosov diffeomorphisms that admit σ -finite SRB measures in spaces with dimensions bigger than one; there exists a C 1+1 ( = C2 , the derivative is Lipschitz continuous) almost Anosov diffeomorphism with a σ -finite SRB measure on a two-dimensional manifold.

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X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

Corollary 2.2. Under the assumptions of Theorem 2.1, if f admits an SRB measure μ, then for any continuous function φ : M → R, one has

lim

n→∞

 n−1 1 φ ( f i (x )) = n

φ dμ, μ-a.e. x ∈ M,

Proposition 3.1.

i=0

where the full μ-measure set depends on each continuous function φ (Birkhoff Ergodic Theorem); if f has a σ -finite SRB measure, then for ν -a.e. x ∈ M, one has n−1 1 δ f i (x) → δ p as n → ∞, n

where δ z is the Dirac measure at z and ν is the Riemannian measure on M, the above convergence is in the weak star topology. For the map f and its invariant probability measure μ, the correlation function for two observables ,  : M → R is defined by



(a) For any x ∈ M, the maps x → Exu and x → Exs are continuous. (b) There exist two continuous foliations F u and F s on M tangential to Eu and Es , respectively, such that (1) the stable manifold at x is given by the stable leaf:

F s (x ) = {y ∈ M : d ( f k (x ), f k (y )) ≤ Cs (κ s )k d (x, y ),

i=0

Corn (,  ; f, μ ) :=

Note that the notations Eu and F u are used by some abuse of notations, instead of Ecu and F cu , to denote a subbundle and foliation that is not uniformly expanding.

( ◦ ( f n ))dμ −



dμ



 d μ,

where Cs is a positive constant; (2) the unstable manifold at x is given by the unstable leaf:

F u (x ) = {y ∈ M : lim d ( f −k (x ), f −k (y )) = 0}; k→∞

(3) there exist two positive constants β and Cu such that Fβu (x ) is the component of F u (x ) ∩ expx (Ex (β )) containing x, and u exp−1 x (Fβ (x )) can be represented by the graph of a func-

tion φxu : Exu (β ) → Exs (β ) with φxu (0 ) = 0 and φxu C 1 ≤ Cu , where exp is the exponential map, and φxu C 1 = supv∈Exu (β ) (|φxu (v )|r + |(φxu ) (v )|r ). Similar results also hold for Fβs (x ).

where n is a positive integer. We obtain the following results by showing the hypotheses of Theorem 6.3 in [21]. Theorem 2.2. Let f be a diffeomorphism satisfying the above assumptions. For any α < m − 1, m − 1 − α < θ ≤ 1, any neighborhood V of p, and any Hölder functions ,  with exponent θ , supp, supp ⊂ MV, and ∫dμ∫ dμ = 0, we have

A (,  ) n

m−1

α −1

≤ |Corn (,  ; f, μ )| ≤

A(,  ) n

m−1

α −1

,

(2.3)

where μ is an SRB measure specified in Theorem 2.1, A (,  ) and A(,  ) are positive constants determined by  and  . 3. The existence of SRB measures This section is devoted to the proof of Theorem 2.1. The conditions for the existence of the finite or σ -finite SRB measures are obtained, the relationship among the measures, the differentiability, and the dimension of the spaces are discussed, where these three objects are linked together by the level sets with the same first return time. This section is divided into three parts. In Section 3.1, the properties of the stable/unstable manfilds are studied. In Section 3.2, the distortion estimates are considered. In Section 3.3, the finite or σ -finite SRB measures are constructed. 3.1. Properties of the stable and unstable manifolds

(β ) := {v ∈

Exu

:

|v|r ≤ β},

and

Ex (β ) := Exu (β ) × Exs (β ).

Exs

(β ) := {v ∈

Exs

:

Proof. Part (a) can inf{κ u (x ) : x ∈ M}. Part (1) of (b) rem 5.5]. Part (2) of (b) is orem 5.5]. Part (3) of (b) rem 5A.1]. 

be derived from the gap assumption κ s < 1 = follows from hypothesis (2) and [24, Theoobtained by using hypothesis (2) and [24, Theis derived by hypothesis (2) and [24, Theo-

For convenience, denote by W s (x ) = F s (x ) (W u (x ) = F u (x )) and Wβs (x ) = Fβs (x ) (Wβu (x ) = Fβu (x )) the stable (unstable) mani-

fold and local stable (unstable) manifold at x, respectively. For any y ∈ Ws (x) (Wu (x)), denote by ds (x, y) (du (x, y)) the minimal distance from x to y on the stable (unstable) manifold Ws (x) (Wu (x)), where the metric on Ws (x) (Wu (x)) is induced by the Riemannian metric restricted to Ws (x) (Wu (x)). By Proposition 3.1, one has the following result immediately. Corollary 3.1. The map f has a local product structure, that is, there are constants 0 < < β such that for any y, z ∈ M with d(y, z) < , one has that [y, z] = Wβu (y ) ∩ Wβs (z ) and [z, y] = Wβu (z ) ∩ Wβs (y ) contain exactly one point, respectively.

In this section, the properties of the stable and unstable manifolds are studied. Assume that f satisfies the hypotheses of Theorem 2.1 throughout this section. For any real numbers A and B, or two real functions A, B : N → R, where the number n ∈ N represents the number of iteration, the notation A ≈ B means that there are two positive constants C0 and C0 such that C0 B ≤ A ≤ C0 B, where A and B might be dependent on the number of iteration, but C0 and C0 do not depend on the number of iteration (This means that C0 and C0 are the same for any fn ). For the Riemannian manifold M, let d( · , · ) be the distance function derived from the Riemannian metric. For any x ∈ M, let Exu and Exs be the unstable and stable tangent spaces at x, respectively. For any positive constant β , set

Exu

∀k ≥ 0},

|v|r ≤ β},

Definition 3.1 [41]. Given a set X in M, if for any x, y ∈ X, one has that [x, y], [y, x] ∈ X, then X is said to be a rectangle, where [x, y] = Wβu (x ) ∩ Wβs (y ). A rectangle P is called proper if intP = P . By Corollary 3.1, it is reasonable to define the following rectangle

[γ s , γ u ] = {Wβu (x ) ∩ Wβs (y ) : x ∈ γ s , y ∈ γ u }, where γ s and γ u are stable and unstable manifolds with sufficiently small diameter, respectively. In the following discussions, assume that the rectangles are defined by [γ s , γ u ]. For any subset X, set W u (x, X ) := Wβu (x ) ∩ X and W s (x, X ) := Wβs (x ) ∩ X. Given two rectangles X1 and X2 , if for any x ∈ X1 with fk (x) ∈ X2 for some k ≥ 0, one has that f k (W u (x, X1 )) ∩ X2 = W u ( f k (x ), X2 ), then it is said that fk (X1 ) u-crosses X2 ; if for any x ∈ X1 with f −k (x ) ∈ X2 for some k ≥ 0, one has that f −k (W s (x, X1 )) ∩ X2 = W s ( f −k (x ), X2 ), then it is said that f −k (X1 ) s-crosses X2 .

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

Definition 3.2 [41]. A Markov partition of M is a finite covering P = {P0 , P1 , . . . , Pl } of M by proper rectangles satisfying that (i) intPi ∩ intPj = ∅ for i = j; (ii) if x ∈ intPi and f(x) ∈ intPj , then f(Wu (x, Pi ))⊃Wu (f(x), Pj ) and f(Ws (x, Pi )) ⊂ Ws (f(x), Pj ). By hypothesis (1) and the existence of Markov partitions for Anosov diffeomorphisms, there exists a Markov partition for the map f. Suppose the Markov partition is

P = {P, P1 , P2 ..., Pl } with p ∈ intP.

(3.1)

For convenience, we assume that the diameter of any element in this Markov partition is sufficiently small. For the existence of Markov partitions with arbitrarily small diameter, please refer to [15]. Definition 3.3 [27]. Let W1 and W2 be two unstable manifolds, and the holonomy map H: W1 → W2 be the continuous map defined by the sliding map along the stable manifolds, that is, H (x ) = W s (x ) ∩ W2 for x ∈ W1 . If H is Lipschitz for every (W1 , W2 , H), then the stable manifold Ws is said to be Lipschitz. Remark 3.1. Note that the holonomy map is well defined if local (not global) stable manifolds are considered. Proposition 3.2. The stable manifold Ws is Lipschitz, that is, for any δ > 0, there is CL > 0 such that given any (W1 , W2 , H) with ds (x, H(x)) < δ for any x ∈ W1 , H is Lipschitz with Lipschitz constant less than CL . Further, if m ≥ 3, then the holonomy map is differentiable. Proof. First, we consider the case m = 2 and 0 < α ≤ 1. By hypothesis (1) , f is C 1+α . We will apply the arguments used in the proof of [27, Proposition 2.5] to obtain the Lipschitz property of the stable manifold Ws . In the arguments of [27, Proposition 2.5], the maps are C2 , whereas the maps considered here are C 1+α . For convenience, we only give the details where the assumption that C 1+α (0 < α ≤ 1) is required. Take a segment γ ⊂ W1 with sufficiently short length. By Proposition 3.1, for any x ∈ γ and any n > 0, we have d ( f n (x ), f n (H (x ))) ≤ Cs κsn d (x, H (x )). Take n large enough such that the length of fn (γ ) and d(fn (x), fn (H(x)) is the same up to a constant, that is, length(fn (γ )) ≈ d(fn (x), fn (H(x)). So, for any y, z ∈ γ , we have

| |

| | ≈ | |

D fyn Eyu D fzn Ezu

n −1

(1 ± const · d ( f (y ), f (z ))α ) i

s

|D fx−1 v|r = 1, |v|r x∈M v∈Exu ,v =0

|D fw−1 |Ewu |d = det(D fw−1 |Ewu ) D fw−1

(3.2) u, Ew

the determinant of restricted to and the value of the determinant might be positive or negative. Since we only consider the points in the same stable or unstable manifolds in the distortion estimates, we can assume that |D fw−1 |Ewu |d is positive for convenience in the following distortion estimates in Propositions 3.3 and 3.4, and Lemmas 3.2 and 3.3. Lemma 3.1 [40, Theorem 9.2]. If a function φ (x ) : S ⊂ Rm → R has a continuous derivative Dφ (x) at all points of the set S, where S is convex, then for any x0 , x ∈ S, the formula holds:

φ ( x ) − φ ( x0 ) =



1 0

Dφ (x0 + t (x − x0 )) · (x − x0 )dt,

where Dφ (x ) = ( ∂∂φ , · · · , ∂∂φ ) and Dφ (x0 + t (x − x0 )) · (x − x0 ) is x1 xm the dot product. Proposition 3.3. Given any small rectangle P, whose interior contains p, there are two positive constants δ and D such that if  is a disk contained in some unstable manifold with diam() ≤ δ and  ∩ P = ∅, then for any y, z ∈  and k > 0, we have

D−1 ≤

|D fy−k |Eyu |d ≤ D. |D fz−k |Ezu |d

(3.3)

0

Corollary 3.1, and δ 0 is sufficiently small so that P ∪ f(P) ⊂ U and diam(P ∪ f(P)) < , where U is given in hypothesis (4) and is specified in Corollary 3.1. So, the stable holonomy map H is well defined with respect to the local unstable manifolds contained in U by Corollary 3.1, that is, the holonomy map H : Wδu (x ) → Wδu (y ) is 0

a = |D f −1 |E u |o = sup sup

0

well defined for all x, y ∈ P such that the respective unstable manifolds satisfy the conclusions of Corollary 3.1. By hypothesis (4), we do not need to consider the curvature on unstable manifolds contained in U.

|D f x v|r = κ s < 1, v =0 |v|r

b = |D f |E s |o = sup sup

Lemma 3.2. Assume m ≥ 3. Let P be a sufficiently small rectangle, whose interior contains p. Given any disk  ⊂ ((f(P)P) ∩ Wu (x)) with x ∈ f(P)P, and f −i () ⊂ P for 1 ≤ i ≤ k − 1, one has

and

x∈M v∈Exu ,v =0

In this section, the distortion estimates are obtained for almost Anosov diffeomorphisms. Denote by

0

Second, we study the cases m = 2 and α > 1, and m ≥ 3. It follows from the assumptions (1)–(5) that we have

c = |D f |E u |o = sup sup

3.2. Distortion estimates

The proof is split into several lemmas. Assume that there is a positive constant δ 0 such that P = [Wδs ( p), Wδu ( p)] by

≤ (1 ± const · length( f (γ )α ))n ≈ (1 ± const · (κ α )n )n ≈ const. n

x∈M v

Remark 3.2. By (2.1), there is a sufficiently small neighborhood U0 of p such that U0 ⊂ U, and the local unstable manifold of any point taken from U0 is contained in some horizontal plane, i.e., the mth coordinate component xm is equal to some constant. This fact will be used in the study of the distortion estimates. We give a proof of this fact. It is evident that this is true for the fixed point p. For any other point, we will show this fact by contradiction. Take two points from one unstable manifold close to the unstable manifold of p, and consider the inverse iteration. Assume that these two points are not taken from a common horizontal plane, then the distance between these points along the vertical axis will become large, this is a contradiction. This implies that the local unstable manifolds are contained in horizontal planes.

i

i=0

∈Exs ,

153

|D f x v|r = L0 > 1. |v|r C2

By hypotheses (1), (4) and (5), we have that the map f is and abc = κ s L0 < 1. By using similar method in the proof of [23, Theorem 6.3], we can show that the holonomy map is differentiable. Hence, the stable manifold Ws is Lipschitz. This completes the proof. 

log

|D fy−k |Eyu |d du (y, z ) ≤ E1 , ∀y, z ∈ , −k u χ |D f z |Ez |d

(3.4)

where E1 is a constant depending on the map f and the diameter of P (or Wu (p, P)),

χ=

inf

w∈∂ s P ∩W u ( p,P )

du (w, f (w )),

(3.5)

154

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

∂ s P = {w ∈ P : w ∈ intW u (w, P )}, du (y, z) is the distance between y and z in the same unstable manifold of U, and the Riemannian metric is used. Proof. It is evident that χ > 0 and χ is dependent on the choice of P, since w and f(w) are in Wu (p). The positive constant E1 will be determined later. It is sufficient to consider the case m = 3, a similar argument also works for m > 3. By Remark 3.2, one has  ⊂ {(x1 , x2 , x3 ) : x3 = E2 }, where E2 is a real number. Set i := f −i (), 0 ≤ i ≤ k. Hence,

i ⊂ Ai := {(x1 , x2 , x3 ) : x3 = κs−i E2 }, 0 ≤ i ≤ k.

(3.6)

The proof is divided into four steps. Step 1. Define a function φ (w ) = |D fw−1 |Ewu |d . For any k ∈ R, a level curve is the collection of all the points w such that φ (w ) = k. We will show that the images of level curves of φ under f are also level curves of φ . By (2.1), set r := x21 + x22 , the map f restricted to the unstable manifold is rewritten as α

Consider the function φ (w ) = |D fw−1 |Ewu |d , w ∈ M. Since f is C2

α

((1 + r 2 )x1 + ψ1 (x1 , x2 , x3 ), (1 + r 2 )x2 + ψ2 (x1 , x2 , x3 )), where ψ1 (x1 , x2 , x3 ) = ρ x23 x1 and ψ2 (x1 , x2 , x3 ) = ρ x23 x2 . By direct calculation, the Jacobian of f with respect to x1 and x2 is



α α 1 1 + r 2 + α r 2 −1 x21 + ∂ψ ∂x α −1

αr 2

2 x1 x2 + ∂ψ ∂ x1

1

1 α r 2 −1 x1 x2 + ∂ψ ∂ x2 α α −1 2 1 + r 2 + α r 2 x2 +



α

∂ψ2 ∂ x2

.

So, the determinant is



∂ψ1 ∂ψ2 α α 1 + r 2 + α r 2 −1 x22 + ∂ x1 ∂ x2 α

α

∂ψ ∂ψ 1 2 α r 2 −1 x1 x2 + − α r 2 −1 x1 x2 + ∂ x2 ∂ x1 ∂ψ α α α 1 = 1 + ( 2 + α )r 2 + ( 1 + α )r α + (1 + r 2 + α r 2 −1 x22 ) ∂ x1 ∂ψ2 α α + (1 + r 2 + α r 2 −1 x21 ) ∂ x2 ∂ψ1 ∂ψ2 ∂ψ2 ∂ψ1 ∂ψ1 ∂ψ2 α α + − α r 2 −1 x1 x2 − α r 2 −1 x1 x2 − ∂ x1 ∂ x2 ∂ x1 ∂ x2 ∂ x2 ∂ x1 α α 2 = 1 + ( 2 + α )r + ( 1 + α )r α α + ρ x23 (2 + 2r 2 + α r 2 ) + ρ 2 x43 . (3.7) α

α

1 + r 2 + α r 2 −1 x21 +

Hence, for fixed x3 , the level curves of the function φ (w) are circles contained in some horizontal plane. This, together with (2.1) and (3.6), yields that the images of level curves under f are also level curves. Step 2. We will show an estimate on the left hand side of Eq. (3.4) by the distance between the backward orbits, that is, we will verify the inequality (3.10) introduced later. Put yi = f −i (y ) and zi = f −i (z ), i ≥ 0. Let O1 be the plane containing the x3 -axis and the point y, O2 be the plane containing the x3 -axis and the point z. It follows from (2.1) that yi ∈ O1 and zi ∈ O2 , 0 ≤ i ≤ k. Denote by lzi the level curves contained in Ai , where each Ai is specified in (3.6). The set lzi ∩ O1 has two points, pick zi∗ ∈ lzi ∩ O1 , which is closer to the point yi . Let Si be the line segment joining the points zi∗ and yi in the plane Ai . An illustration diagram for m = 3 is provided in Fig. 1. By (2.1) and our construction, one has that Si+1 = f −1 (Si ), and there is a line segment  0 such that S0 ⊂  0 and the endpoints of  0 are two points w and w with w ∈ ∂ s P and w ∈ ∂ s f(P), where ∂ s P is introduced in (3.5), and ∂ s f(P) is defined similarly. Set i := f −i (0 ), i ≥ 0. So,

length(0 ) ≥ C0 χ and Si ⊂ i ,

Fig. 1. The illustration graph of the discussions in Step 2 for m = 3, where the plane O1 is in green color, the plane O2 is cyan color, lzi is in blue color, and Si is in red color (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

(3.8)

where C0 is a positive constant determined by Proposition 3.2.

by hypothesis (1), f −1 is C2 by the Inverse Mapping Theorem and D fw−1 is C1 . This, together with Part (3) of (b) of Proposition 3.1, implies that D fw−1 |Ewu is C1 . By (3.2) and the fact that the dimen-

sion of M is m, the function φ (w ) = |D fw−1 |Ewu |d could be thought of as a differentiable function from Rm to R for convenience. By the discussions above, the point zi∗ lies in a uniformly small neighborhood of the point yi for sufficiently large i and sufficiently small . So, |D fw−1 |Ewu |d can be defined on a small convex neighborhood of yi , where this neighborhood contains both the points zi∗ and yi . Thus, Lemma 3.1 could be applied here, and one has

 −1  |D fy |Eyu |d − |D fz−1  ∗ |E u∗ |d i z i i i  1    ∗ u =  D(|D f (−1 | | )( y − z ) dt  ∗ ∗ i i z +t (yi −z )) E(z∗ +t (y −z∗ )) d i

0

i

i

i

i

≤ C1 |D(|D fy−1 |Eyu |d )du (yi , zi∗ ) ≤ C2 du (yi , zi∗ ), i i

(3.9)

where C1 and C2 are two positive constants. Hence, for any j ≤ k, by hypothesis (1), one has

log

|D fy− j |Eyu |d |D fz− j |Ezu |d

≤ C3



1+

  |D fy−1 |Eyu |d − |D fz−1 |Ezu |d  i i i i |D fz−1 |Ezui |d i

i=0

j−1   i=0

= C3

≤ log

j−1

 |D fy−1 |Eyu |d − |D fz−1 |Ezu |d  i i i i

j−1   i=0

j−1   |D fy−1 |Eyu |d − |D fz−1  du (yi , zi∗ ), ∗ |E u∗ |d ≤ C3C2 i z i i i

(3.10)

i=0

where C3 > 0 is a constant depending on f. Step 3. We will give an estimate on du (yi , zi∗ ) in (3.10). Recall that  i is a line segment, for any w ∈  i , denote by |D f −1 |i (w) |o = |D f −1 (w )vw |e the length of the vector D f −1 (w )vw , where vw is a tangent vector with unit length at the point w. Note that |D f −1 |i (w ) |o could be thought of as the contraction rate along the direction of the line segment  i . It follows from the twicedifferentiability of f and Lemma 3.1 that

 −1  |D f | (y ) |o − |D f −1 | (z∗ ) |o i i i i  1    =  D(|D f −1 |i (zi∗ +t (yi −zi∗ )) |o )(yi − zi∗ )dt  0

≤ C4 |D(|D f −1 |i (yi ) |o )|du (yi , zi∗ ) ≤ C5 du (yi , zi∗ ) ≤ C5 length(i ),

(3.11)

where C4 and C5 are two positive constants. Hence, for any j ≤ k, by (3.11),

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

   j−1 |D f −1 | (y ) |o − |D f −1 | (z∗ ) |o |D f − j |0 (y) |o i i i i log ≤ log 1+ |D f −1 |i (zi∗ ) |o |D f − j |0 (z∗ ) |o i=0 ≤ C6

j−1  

j−1   |D f −1 | (y ) |o − |D f −1 | (z∗ ) |o ≤ C6C5 du (yi , zi∗ ), i i i i

i=0

i=0

where |D f −1 |i (z∗ ) |o is uniformly bounded away from zero, and i

C6 > 0 is a constant depending on f. Thus, it follows from (3.12) that

length( j ) where C7

≤ C7

du (y, z∗ ) , length(0 )

∀ j ≤ k,

is a positive constant depending on f. An in-

tuitive understanding (rough idea) of (3.13) is |D f − j |

(3.13)

u ∗ (y ) |o d (y,z )

d u (y

∗ j ,z j )

length( j )

≈

du (y,z∗ )

≈ , where the mean value thelength(0 ) length(0 ) orem is used for some abuse of notations. The inequality (3.13) means that the ratio of a subsegment and the whole segment contained in some unstable manifold are controlled under the backwards iterates. Step 4. We show (3.4). Recall that H is the holonomy map from i to H(i ) ⊂ Wu (p). ˆ i be the images of  i under the map H, 0 ≤ i ≤ k. In fact, the Let  ˆ i are pairwise disjoint by our construction, implying that subsets  |D f − j |

j−1 

0

∗ |o 0 (z )

length(i ) ≤

i=1

j−1 

CL length(ˆ i ) ≤ CL diam(W u ( p, P )),

(3.14)

log

|D fz− j |Ezu |d

≤

C3C2 d ( u

yi , zi∗

du (y, z∗ )

χ

≤ E1

)

where E4 is a positive constant, and the first inequality is derived by the fact that  is homeomorphic to an interval and zi ∈ [yi , yi+1 ] by abuse of notations. This, together with Lemma 3.3 in [26], yields that (3.16) holds.  Now, we are ready to prove Proposition 3.3. Proof. First, we show (3.3) for the case m = 3 and α satisfying hypothesis (4). The same arguments for m = 3 also work for m > 3 and α satisfying hypothesis (4), and m = 2 and α > 1. By the properties of the Markov partition, there is a sufficiently small constant δ > 0 such that for any disk  with diameter less than δ and contained in some unstable manifold, and i = f −i (), i ≥ 0, we have (i) i ⊂ Pl0 for some 0 ≤ l0 ≤ l, l0 is dependent on i; (ii) if int(i ) ∩ (f(P)P) = ∅, then i ⊂ f(P)P. Suppose that the number of the passages through P up to time k for the orbits of y and z is s0 , and there exist non-negative integers ki and li with ki < ki + li < ki+1 , 1 ≤ i ≤ s0 , such that

∀j ∈

P ∩  j = ∅,



((ki , ki + li ) ∩ Z ),

du (y, z )

χ

(3.15)

Lemma 3.3. Assume m = 2 and 0 < α ≤ 1. Let P be a sufficiently small rectangle with interior containing p. If  is homeomorphic to an interval,  ⊂ ((f(P)P) ∩ Wu (x)) for some x ∈ f(P)P, diam() is small enough, and f −i () ⊂ P for 1 ≤ i ≤ k − 1, then

(3.16)

α . where D is a positive constant and ϑ ∗ = 1+ α

Proof. By Proposition 3.2, the stable manifold Ws is Lipschitz, so it suffices to study the distortion estimate on the unstable manifold of the indifferent fixed point p. It is evident that f is injective when it is restricted to Wu (p, P) and f −1 (W u ( p, P )) ⊂ W u ( p, P ). For convenience, when f is restricted to the unstable manifold of α + ψ (x ) for x > 0, where p, suppose p = 0 and f (x1 ) = x1 + x1+ 1 1 1 ψ (x1 ) is the higher order term. Assume that  ⊂ f(Wu (p, P))Wu (p, P). Hence, for any y, z ∈  with du (y, z) ≤ |y|/2, one has

|D f ( y )| log ≤ E3 |y|α −1 du (y, z ), |D f ( z )|



((ki , ki + li ) ∩ Z ),

ki

(3.17)



du ( f (y ), f (z )) ≥ (1 + E2 |y|α )du (y, z ),

∀ j ∈

i

,

|D fz−k |Ezu |d

∗ ≤ D du (y, z )ϑ , ∀y, z ∈ , |D fy−k |Eyu |d

and

s0 s0 ki +1 −1 |D fy−lki i |Eyuk |d  |D fy−1 |Eyuj |d |D fy−k |Eyu |d  j i log = log + log , −1 u −k u −li |D f z j |Ez j |d |D fz |Ez |d i=1 |D fzk |Ezu |d i=0 j=ki +li

du (y, z∗ ) length(0 )

E1 = C3C2C7CL diam(W u ( p, P ))/C0 .

log

∗

where  j = f − j (). So,

where

This verifies (3.4).

∗

1≤i≤s0

i=0

≤ C3C2C7CL diam(W u ( p, P )) ≤ E1

∗

P ∩  j = ∅,

where Proposition 3.2 is used. Hence, it follows from (3.8), (3.10), (3.13), and (3.14) that j−1 

du (yi , zi )1−ϑ ≤ du (yi , yi+1 )1−ϑ ≤ E4 |yi + yαi +1 − yi |1−ϑ

1≤i≤s0

i=1

|D fy− j |Eyu |d

where E2 and E3 are two positive constants. For any y, z ∈ , put yi = f −i (y ) and zi = f −i (z ). Direct calculation gives us ∗ = E4 |yi |(1+α )(1−ϑ ) = E4 |yi |,

(3.12)

du (y j , z∗j )

155

where k0 = l0 = 0 and ks0 +1 = k + 1. The first part of the right hand side of (3.17) is estimated by (3.4), the first and second parts of the right hand side of (3.17) are bounded by a geometric sequence, since the corresponding part of the orbit lies in MP, and f is uniformly hyperbolic on MP by hypothesis (3). Hence,

log

s0 s0 |D fy−k |Eyu |d  d u ( yki , zki )  ≤ E + E5 d u ( yki +li , zki +li ) 1 χ |D fz−k |Ezu |d i=1 i=0 ≤ E6 du (y, z ),

(3.18)

where E5 and E6 are two positive constants. So, taking D = E6 δ, by the choice of the δ as above, one has that (3.3) holds. Second, we study (3.3) for the case m = 2 and 0 < α ≤ 1. By using (3.16) in Lemma 3.3 and the similar argument in (3.17) as above, we have that (3.3) holds for the case m = 2 and 0 < α ≤ 1. This completes the proof.  Remark 3.3. We sketch a proof of the claim that E1 /χ in (3.4) goes to positive infinity as diam(P) goes to zero. This implies that the constant D in (3.3) is dependent on the choice of P. By (2.1), it is enough to consider m = 2 and use the map h(x ) = x + x1+α for x > 0 to represent the dynamics of f restricted to Wu (p). By the definition of χ in (3.5) and the definition of E1 in (3.15), and 1 1 Lemma 3.5, we have that E1 ≈ 11 and χ ≈ 1 − 1 . So, (k−1 ) α

kα

E1

χ

1

≈

1

kα 1 1

(k−1 ) α

−

1 1

kα

=

1

( k−1 )− α − 1 k 1

≈ kα ,

kα

156

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

which approaches to positive infinity as the diameter of Wu (p, P) goes to zero, since the diameter of Wu (p, P) is about 11 . kα

Proposition 3.4. There exists a positive constant δ such that if  is a disk contained in some stable manifold with diam() ≤ δ , then for any y, z ∈  and k > 0, we have

log

|D fyk |Eyu |d ≤ D¯ (ds (y, z ))α , if m = 2 and 0 < α ≤ 1; |D fzk |Ezu |d

log

|D fyk |Eyu |d ≤ D ds (y, z ), if m ≥ 3, or m = 2 and α > 1, |D fzk |Ezu |d

where D¯ and D are two positive constants. Proof. The proof consists of two steps. First, we study the situation m ≥ 3 and α satisfying (4). Second, we investigate m = 2 and 0 < α ≤ 1. The statement for m = 2 and α > 1 could be derived by using the arguments for m ≥ 3. Step 1. We study the situation that m ≥ 3 and α satisfying (4). Suppose the diameter of  is small enough. Put yi = f i (y ) and zi = f i (z ), i ≥ 0. By hypothesis (1), f is C2 . So the function φ (x ) = |D fx |Exu |d is uniformly continuous and differentiable. This, together with Lemma 3.1, yields that

  |D fyi |Eyu |d − |D fzi |Ezu |d  i i  1    =  D(|D f (zi +t (yi −zi )) |E(uz +t (y −z )) |d )(yi − zi )dt  i

0

i

i

≤ C¯1 |D(|D fyi |Eyu |d )|ds (yi , zi ) ≤ C¯2 ds (yi , zi ), i

where C¯1 and C¯2 are two positive constants, and ds (yi , zi ) → 0 as i → +∞. Hence, for any j ≤ k,

log

|D fyj |Eyu |d |D fzj |Ezu |d

≤ log

j−1



1+

  |D fyi |Eyu |d − |D fzi |Ezu |d  i

|D fzi |Ezui |d

i=0

≤ C¯3

i

j−1  

 |D fyi |Eyu |d − |D fzi |Ezu |d  i i

i=0

j−1 

≤ C¯3C¯2

d s ( yi , zi )

where C¯5 > 0 is a constant depending on f, and Cs is introduced in (b) of Proposition 3.1. This completes the proof.  Proposition 3.5. The unstable manifold Wu (p) and the stable manifold Ws (p) are dense in M, respectively. Proof. We only show that Wu (p) is dense. Similar arguments work for Ws (p). Take any rectangle X with intX = ∅ and a strictly smaller rectangle Xˆ ⊂ intX. It follows from hypothesis (1) that there is k > 0 such that f −k (Xˆ ) ∩ P = ∅. So, if k is sufficiently large, then f −k (X ) scrosses P. Hence, fk (Wu (p, P)) ∩ X = ∅. This completes the proof.  Assume P = [Wδu ( p), Wδs ( p)], where δ 0 is a small positive 0

Lemma 3.4. There exists an ergodic invariant Borel probability measure ν g for the map g, which has absolutely continuous conditional measures on unstable manifolds of f. Proof. First of all, we show the existence of an invariant measure. Let Pˆ be one of the connected components of f(P)P. Set Q := W u (x, Pˆ ) for some x ∈ Pˆ. Denote by ν Q the induced Riemannian measure on Q, and define gk∗ νQ as the push-forward of ν Q , that is, (gk∗ νQ )(E ) = νQ (g−k (E )) for any measurable set E. Take a limit of  −1 i the sequence 1k ki=0 g∗ νQ in the weak star topology, denoted by ν g . It is evident that ν g is an invariant measure with respect to g. Second, we show that ν g has absolutely continuous conditional measures on unstable manifolds with respect to the induced Riemannian measures. For any small rectangle K in MP, each component of gi (Q) is a disjoint union of Wu leaves, which are contained in some element in the Markov partition, and if any component of gi (Q) intersects K, then it u-crosses K by hypothesis (1) and the discussions used in the proof of Proposition 3.5. Let ρ i be the density of gi∗ νQ with respect to the induced Riemannian measure on gi (Q). It follows from Proposition 3.3 that for any x, y in the same component of gi (Q) ∩ K,

i=0

≤ C¯3C¯2Cs

j−1 

κ

i s sd

(y, z ) = D ds (y, z ),

i=0

where C¯3 > 0 is a constant depending on f, Cs is specified in (b) of Propositions 3.1 and 3.1 is used. Step 2. Let us consider the case m = 2 and 0 < α ≤ 1. Taking yi = f i (y ) and zi = f i (z ), i ≥ 0. By hypothesis (1) , f is C 1+α , implying that

  |D fyi |Eyu |d − |D fzi |Ezu |d  ≤ C¯4 (ds (yi , zi ))α , i i

where C¯4 is a positive constant, and ds (yi , zi ) → 0 as i → +∞. Hence, for any j ≤ k, by (b) of Proposition 3.1,

log

|D fyj |Eyu |d |D fzj |Ezu |d

≤ log

j−1

  |D fyi |Eyu |d − |D fzi |Ezu |d   i i 1+ |D fzi |Ezui |d



i=0

≤ C¯5

j−1   i=0

≤ C¯5C¯4

 |D fyi |Eyu |d − |D fzi |Ezu |d  i i

j−1 

(d (yi , zi ))α s

i=0

≤ C¯5C¯4Csα

j−1  i=0

κsαi (ds (y, z ))α = D¯ (ds (y, z ))α ,

0

constant. Define the first return map g = f τ (x ) (x ) : M \ P → M \ P, where τ (x ) = min{i > 0 : f i (x ) ∈ M \ P } is the first return time function with respect to the set MP.

D−1 ≤

ρi ( x ) ≤ D, ρi ( y )

where D is specified in (3.3), and is independent of i. Similar estimates also hold for the limit densities by letting i go to infinity. Hence, ν g has absolutely continuous conditional measures on unstable manifolds. Finally, the ergodicity of g with respect to ν g is derived by using the Hopf argument and similar arguments in the proof of [27, Lemma 5.3], where the application of [27, Lemma 5.1] in the proof of [27, Lemma 5.3] is replaced by the existence of Markov partition in hypothesis (1). This completes the proof.  3.3. Construction of SRB measures In this section, the finite or σ -finite SRB measures are constructed. Define

S = f −1 (P ) \ P and S(k ) = {x ∈ S :

τ ( x ) ≥ k}, k ≥ 1,

(3.19)

where P is the element in the Markov partition P specified in (3.1). It is evident that

S = S(2) = {x ∈ M \ P,

τ (x ) > 1}, and S(k+1) ⊂ S(k) ∀k ≥ 2.

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

157

tive constants Eα and Eα such that

Eα

Eα

≤ wk ≤

1

kα

1

kα

,

∀k ≥ 1 .

(3.22)

Note that the dimension of the unstable manifold is m − 1. So,



there exist two positive constants Eα and Eα such that



Eα k

m−1

α



Eα u u ≤ νW , u ( p) (γ ) ≤ m−1 k k α



Fig. 2. The illustration graph of S(k) for m = 2, where the stars represent the orbit of a point with first return time three.

∀k ≥ 1 .

(3.23)



Note that Eα and Eα are dependent on the choice of the initial value w0 . Now, we will look at how to choose the initial value w0 , and



show that there are two positive constants Dα and Dα such that they are independent of the choice of w0 and an inequality like (3.20) holds. Since the boundaries of the elements in a Markov paru might tition might not be smooth by Bowen [16], the shape of γ k u be irregular, and the boundary of γ might be very strange. So, k

W s (x, S(k ) )

W s (x, S )

Wu (x,

S(k) ) ⊂ Wu (x,

So, one has that = and S) for any x ∈ S(k) . In Fig. 2, an illustration graph of S(k) on the plane is provided, i.e., m = 2. We introduce the following two lemmas, which will be used to estimate the measure of the set S(k) , that is, (3.20). Lemma 3.5 [25, Lemma 8.1]. Let {ak }∞ be a sequence of positive k=1 numbers, Ci and ηi be positive constants, i = 1, 2. 1+η1

(i) If ak−1 ≥ ak + C1 ak

for any k ≥ 1, then there exist D > 0 and − η1

k0 ≥ 1 such that ak ≤ D(k − k0 ) 1 for sufficiently large k. 1+η (ii) If ak−1 ≤ ak + C2 ak 2 for any k ≥ 1, then there exist D > 0 and k 0 ≥ 1 such that ak ≥ D (k + k 0 )

− η1

2

for sufficiently large k.

Lemma 3.6. Given any positive integer ω and positive constant α c , consider the function h : [−1, 1] → R, which is written as h(x ) = x(1 + xαc + o(xαc )) for x in a small neighborhood I of 0. For any  ω a0 ∈ (0, 1] ∩ I, set ak := h−k (a0 ), k ≥ 1. If 0 < α c < ω, then ∞ k=1 ak < ∞ ω ∞; if α c ≥ ω, then k=1 ak = ∞. Proof. By Lemma 3.5, one has that ak ≈ k−1/αc for sufficiently large ∞ ω ∞ ω k. So, if 0 < α c < ω, then k=1 ak < ∞; if α c ≥ ω , then k=1 ak = ∞.  Proposition 3.6. For any unstable manifold γ u ∈ Wu (S), let νγu be the Riemannian measure induced on γ u . Then,

D α k

m−1

α

≤ νγu (γku ) ≤

Dα k

m−1

α

,

(3.20)

where γku = γ u ∩ S(k ) , and D α > 0 and Dα > 0 are two constants. u be the projection of γ u onto Wu (P) along Ws by Proof. Let γ k k the holonomy map. It follows from Proposition 3.2 that we have u u u u u u νW u ( p) (γk ) ≈ νγ (γk ). So, it is sufficient to estimate νW u ( p) (γk ). By u (2.1), the map f restricted to W (p) is written as

f (x1 , x2 , . . . , xm−1 , 0 ) = (1 + |(x1 , . . . , xm−1 )|αe )(x1 , . . . , xm−1 , 0 ). (3.21) By (3.21), it suffices to consider the case x2 = · · · = xm−1 = 0 and u has a regular shape, e.g. a disk. So, we simplify this x1 > 0, if γ k problem to the study of the map h(x1 ) = (1 + xα )x1 , x1 > 0. For any 1 sufficiently small initial point w0 > 0 (this w0 is taken as the intersection of the boundary of γu and the positive x -axis), consider k

1

the backwards orbit of w0 under h, set wi := h−i (w0 ), i ≥ 0, that is, α , i ≥ 1. And, it is evident that w > w >  → 0, wi−1 = wi + w1+ 0 1 i and h([wi+1 , wi ] ) = [wi , wi−1 ]. By Lemma 3.5, there exist two posi-

u might need any initial point w0 taken from the boundary of γ k





different positive constants Eα ,w0 and Eα ,w0 to make some inequality like (3.23) hold. By Proposition 3.3, there exist two constants



Dα and Dα , which are independent of the choice of w0 , and



Dα k

m−1

α

u u ≤ νW u ( p) (γ ) ≤ k



Dα k

m−1

α

,

∀k ≥ 1 .

This, together with Proposition 3.2, yields that (3.20) holds.



Remark 3.4. The boundaries of the Markov partitions for some hyperbolic toral automorphisms in T2 are smooth [47]. However, the boundaries for some hyperbolic toral automorphisms in T3 are not smooth [16]. In [18], Cawley showed that the only hyperbolic toral automorphisms for which there exist Markov partitions with piecewise smooth boundary are those for which some iteration of the automorphisms is linearly covered by a direct product of automorphisms of the 2-torus. And, the existence of smooth boundaries for Markov partitions for arbitrary Anosov diffeomorphisms is an open problem. Actually, the fractal structure of the boundary of the Markov partitions does not affect the discussions above. First, the measure of the boundaries of the Markov partitions is zero [33, Chapter IV, Theorem 9.6], where the measure is ergodic. Second, we have the distortion estimate (Proposition 3.3). Now, we show Theorem 2.1. Proof. Set Ri := {x ∈ M \ P : τ (x ) = i}. Denote

μ :=

∞  i−1 

f∗j (νg |Ri ),

i=1 j=0

where ν g is the measure obtained in Lemma 3.4. So, μ is invariant and has absolutely continuous conditional measures on unstable manifolds by Lemma 3.4. By (3.19), note that S(k ) = ∪i≥k Ri , and fi (S(i) ) are pairwise disjoint subsets of P. This, together with the fact that μ is invariant, (3.20), and Lemma 3.6, implies that

μ (P ) ≈ ≈

∞  i=1 ∞  i=1

μ( f i (S(i) )) =  i−

m−1

α

∞ 

μ ( S (i ) )

i=1

< ∞, = ∞,

if 0 < α < m − 1 if α ≥ m − 1.

Hence, if 0 < α < m − 1, then μ is an SRB measure (Similar construction of SRB measures could be found in [31]); if α ≥ m − 1, then the measure of μ of the whole manifold is infinite. Further, for any open neighborhood V of p, the set M \ (∩ki=−k f i (P )) contains the set MV for large enough k, and it is a finite set with

158

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

respect to the measure μ. This means that μ is at most σ -finite. Hence, if α ≥ m − 1, then μ is a σ -finite SRB measure. This completes the proof of Theorem 2.1.  Finally, we prove Corollary 2.2.

Proposition 3.7. For the local expression (2.1), if m = 2 and α ∈ (1, +∞ ), then f is C2 ; if m = 2 and α ∈ (0, 1], then f is only differentiable; if m ≥ 3 and α ∈ {2} ∪ (3, +∞ ), then f is C2 . Proof. First of all, let us study the case that m = 2. Suppose f = ( f1 , f2 ) by (2.1). So,

f 1 ( x1 , x2 ) =

α + ρ x2 x , x1 + x1+ 2 1 1 x1 + (−x1 )α x1 + ρ x22 x1 ,

if x1 ≥ 0 if x1 ≤ 0.

Direct calculation tells us that

 ∂ f1 1 + (1 + α )xα1 + ρ x22 , = 1 + (1 + α )(−x1 )α + ρ x22 , ∂ x1

if x1 ≥ 0 if x1 ≤ 0,

which is continuous for α > 0. So, the second derivative is

 ∂ 2 f1 (1 + α )α xα1 −1 , = 2 −(1 + α )α (−x1 )α −1 , ∂ x1

1

∂f

∂2 f

1 uous; if α > 1, then ∂ x1 is continuous; if α > 1, then is con∂ x21 2

∂2 f

Proof. This is derived from the Birkhoff Ergodic Theorem for SRB measures and the arguments used in the proof of [27, Theorem B] for σ -finite SRB measures. 



We only need to study the continuity of these derivatives at the ∂f origin. Direct computation tells us that if α > 0, then ∂ x1 is contin-

if x1 ≥ 0 if x1 ≤ 0,

which is continuous for α > 1 and discontinuous for α ≤ 1. Hence, since partial derivatives with respect to x2 are C∞ , if m = 2 and α ∈ (1, +∞ ), then the local expression for f is C2 ; if m = 2 and α ∈ (0, 1], then f is only differentiable, f is C1 -smooth with a α -Hölder derivative. Finally, let us study the situation that m = 2. By (2.1), it is sufficient to consider the case m = 3. It is evident that if α = 2, then f is C2 . Now, we study the case that α > 3. Suppose f = ( f1 , f2 , f3 ) in (2.1). If

∂ f1 ∂ f1 ∂ f2 ∂ f2 ∂ 2 f1 ∂ 2 f1 ∂ 2 f1 ∂ 2 f1 ∂ 2 f2 , , , , , , , , , 2 ∂ x1 ∂ x2 ∂ x1 ∂ x2 ∂ x1 ∂ x1 ∂ x2 ∂ x2 ∂ x1 ∂ x22 ∂ x21 ∂ 2 f2 ∂ 2 f2 ∂ 2 f2 , , ∂ x1 ∂ x2 ∂ x2 ∂ x1 ∂ x22 all exist and continuous, then f is C2 . By (2.1), it only needs to study the following terms:

∂ f1 ∂ f1 ∂ 2 f1 ∂ 2 f1 ∂ 2 f1 ∂ 2 f1 , , , , , . ∂ x1 ∂ x2 ∂ x21 ∂ x1 ∂ x2 ∂ x2 ∂ x1 ∂ x22 By direct calculation, we have

∂ f1 α α = 1 + (x21 + x22 ) 2 + ρ x23 + α (x21 + x22 ) 2 −1 x21 , ∂ x1 ∂ f1 α = α (x21 + x22 ) 2 −1 x1 x2 , ∂ x2 ∂ 2 f1 α α = 3x1 α (x21 + x22 ) 2 −1 + (α − 2 )α (x21 + x22 ) 2 −2 x31 , ∂ x21 ∂ 2 f1 ∂ 2 f1 α α = = x2 α (x21 + x22 ) 2 −1 + (α − 2 )α (x21 + x22 ) 2 −2 x21 x2 , ∂ x1 ∂ x2 ∂ x2 ∂ x1 ∂ 2 f1 α α = x1 α (x21 + x22 ) 2 −1 + (α − 2 )α (x21 + x22 ) 2 −2 x1 x22 . ∂ x21

∂2 f

tinuous; if α > 3, then ∂ x ∂1x = ∂ x ∂1x is continuous; if α > 3, then 1 2 2 1 ∂ 2 f1 is continuous. Hence, if ∂ x22

This completes the proof.

α > 3, then f is C2 . 

4. Decay of correlations This section is allocated to the proof of Theorem 2.2. This section is split into three sections. In the first section, a quotient (m − 1 )-dimensional almost expanding system with an indifferent fixed point is introduced by considering a Markov partition and collapsing the stable manifolds in each element of the partition. In the second section, the lower and upper polynomial bounds for the decay rates of the correlation functions of observables on the quotient manifold is obtained. In the last section, the polynomial decay rates for the original system is studied by using the estimates for the quotient map. Remark 4.1. The general idea of the estimates of the decay rates of the correlation functions in [21,44,49,50] can be summarized as follows: consider a basic set for a map, where the basic set refers that the map has some kind of hyperbolicity on this set, and define the first return time function on the basic set, that is, the first time of the point comes back to the basic set under forward iteration; the transfer operator is studied, and this operator acts on some observables, e.g., Hölder observables, and this operator is closely related with the correlation function (for example, consider a map f: M → M, a measure μ on M, and Hölder observables   and  , the transfer operator fˆ is defined by M ( ◦ f n ) ·  dμ =   · ( fˆn  )dμ under certain conditions); the measures of the M

subsets with the same first return time provide a lot of information on the transfer operators. For more details on a transfer operator and its application, please refer to [7,45]. 4.1. The quotient map In this section, a quotient map by collapsing the stable manifolds in each element of the Markov partition is introduced. For the given finite Markov partition P = {P, P1 , . . . , Pl } specified in (3.1), without loss of generality, assume V ⊂ intP, where V is specified in Theorem 2.2. If V is “big”, take several elements from the Markov partition, such that V is contained in the union of these elements, where the union of these elements is regarded as P, and the diameter of the elements in this Markov partition is small enough. Given any Pi and x ∈ Pi , 0 ≤ i ≤ l (for convenience, P0 = P for i = 0), let γ s (x) be the connected component of stable manifold containing x in Pi , and W s (Pi ) = ∪x∈Pi γ s (x ) be the set of all such stable manifolds. Define γ u (x) and Wu (Pi ) similarly. We define an equivalence relation on M by x ∼ y, whenever x and y are belong to the same stable manifold γ s ∈ Ws (Pi ) for some Pi , and denote by x¯ = γ s (x ) the equivalence class containing x. Set

M := M/ ∼, P i := Pi / ∼, and P := {P , P 1 , . . . , P l }. There is a natural projection from M to M. Take an arbitrary γˆiu ∈ W u (Pi ), 0 ≤ i ≤ l. Recall that H : Pi → γˆiu is the sliding map along stable manifolds such that H (x ) = xˆ := γ s (x ) ∩ γˆiu , where x ∈ Pi and γ s (x) ∈ Ws (Pi ). Since P is a Markov partition, we have that f(γ s (x)) ⊂ γ s (f(x)) and f(γ u (x))⊃γ u (f(x)) for any x ∈ Pi and f(x) ∈ Pj . So the quotient map f : x¯ ∈ M → f¯(x¯ ) = f (x ) ∈ M is well defined, and P is a Markov partition for f¯. Let B be the completion of the Borel algebra on M.

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

Now, we define a reference measure υ on M. For each γ ∈ Wu (Pi ), denote by ν γ the Riemannian measure restricted to γ . We introduce the following function

uk (x ) :=

k−1  



log(|D fxi |Exu |d ) − log(|D fxi |Exu |d ) , i

i=0

i

where xi = f i (x ) and  xi = f i ( x ). By Proposition 3.4, one has that uk converges uniformly to some function u. The measure υ is defined by dυγ (x ) = eu(x ) dνγ (x ). By using the statement and proof of (1) of Lemma 1 in Section 3.1 in [49], it is reasonable to introduce a measure υ on M satisfying υ|P = υγˆ u . i

i

From the definition above, the Jacobian of f with respect to υ is

Jν ( f )(x ) = |D fx |Exu |d · eu( f (x )) · e−u(x )

(4.1)

for υγ almost every x ∈ M. By applying the statement and proof of (2) of Lemma 1 in Section 3.1 in [49], one has that J ( f )(x¯ ) is equal to J(f)(y) for any y ∈ γ s (x). Let μ be the SRB measure obtained by Theorem 2.1. So μ induces an invariant measure μ on M naturally. By Proposition 3.3, we have the equivalence of the conditional measure of μ and the induced Riemannian measure on any unstable manifold γ u away from the indifferent fixed point p. This tells us that μ is equivalent to υ away from p¯ , and it has an absolutely continuous measure with respect to υ . So, we have the Markov map (M, B , μ ¯ , f , P ) satisfying the properties: (i) (Generator property) the complete and smallest σ -algebra −k

containing ∪k≥0 f (P ) is B , where the completion of a σ algebra is the one containing all the subsets with measure zero; (ii) (Markov property) f (P i ) ⊃ P j (mod ν¯ ), whenever μ ¯ ( f (P i ) ∩ P j ) > 0 for any P i , P j ∈ P ; (iii) (Local invertibility) the map f : P i → f (P i ) is invertible with measurable inverse for any P i ∈ P with μ ¯ ( P i ) > 0. For more details on Markov map, please refer to [1,44]. Definition 4.1. Consider a general Markov map (X, B, m, T , P ). A Markov map T is called irreducible if for any Pi , Pj ∈ P, there is a positive integer n such that m(T −n (Pi ) ∩ Pj ) > 0. The Markov map is said to be topologically mixing if for any Pi , Pj ∈ P, there is an integer N0 > 0 so that for any n ≥ N0 , we have that Pj ⊂ Tn (Pi ). For any observables ,  : X → R, the transfer operator Tˆ associated to T is defined by



X

( ◦ T ) ·  dm =



X

 · (Tˆ  )dm,

which is represented as

Tˆ (x ) =

 T ( y )= x

dm gm (y )(y ) and gm = . dm ◦ T

The variations of  are defined by

Varn () = sup{|(x ) − (y )| : x, y ∈ [P0 , . . . , Pn−1 ] −1 −i = ∩ni=0 T (Pi )}.

Let be a subset of the partition P. Set Y := ∪P∈P ∗ P, the induced map TP ∗ : Y → Y is the first return map from Y to Y, that is, TP ∗ = T τ . A measure mP ∗ defined by mP ∗ = m|Y is invariant with respect to TP ∗ . Consider a partition of Y: 0

1

n−1 , Y ]

=(

−2 −i ∩ni=0 TP ∗

If P ∗ is finite, then TP ∗ has the “big image” property. Set

gmP ∗ :=

dmP ∗ . dmP ∗ ◦ TP ∗

It is said that log gmP ∗ is locally Hölder continuous, if there exist constants C > 0 and 0 < θ < 1 such that for any n ≥ 1, one has Varn (log gmP ∗ ) ≤ C θ n , where Varn is the variation with respect to the induced map TP ∗ . The element for this partition is denoted by [d0 , . . . , dn−1 ]P ∗ , where d0 , . . . , dn−1 ∈ P ∗ . For any x, y ∈ Y, set

s(x, y ) := sup{n : x, y ∈ [d0 , . . . , dn−1 ]P ∗ }. For any  : Y → R, set

||s := sup

x,y∈Y

|(x ) − (y )| , ϑ s(x,y)

where ϑ ∈ (0, 1) is a constant. Let L be the space of functions  : Y → R with the norm L = ∞ + ||s . Now, we introduce a very useful result. Theorem 4.1 [21, Theorem 6.3]. Let (X, B, m, T , P ) be a topologically mixing probability preserving Markov map, P ∗ be a subset of P, and Y = ∪P∈P ∗ P . Assume that TP ∗ has big image property and log gmP ∗ is locally Hólder continuous. Assume m({x ∈ Y : τ (x ) > n} ) = O(1/nη ) for some η > 1, where τ is the first return time from Y to Y. Then there exists C > 0 such that for any  and  with support inside Y,

  Corn (,  ; T , m ) −



∞ 



m ( {x ∈ Y :

τ ( x ) > k} )

(P )) ∩ T −(n−1) (Y ) i

P∗

= ∅ : P0 ∈ P ∗ , Pi ∈

P ∗ , 1 ≤ i ≤ n − 1}.







  

k=n+1

≤ CFη (n ) ∞ L , where

Fη (n ) =



1/n2η−2 , log n/n2 , 1/nη ,

if 1 < η < 2 if η = 2 if η > 2.

4.2. Polynomial decay rates for the quotient maps In this section, the lower and upper polynomial bounds for the decay rates for the quotient map are obtained.  := M \ P . Recall Now we define the induced Markov map. Set M τ that g = f is the first return map on MP. It is evident that g on  to itself, denoted by  MP induces a first return map from M f . It  follows from p ∈ int P that p¯ ∈ int P . A measure μ ¯M  is defined on M by μ ¯M = μ| ¯ .   M . Set T := Tˆ ∨ Let P 0 = P \ {P } be the Markov partition of M  : τ (x¯ ) = k} : k = 1, 2, . . .} is a partition P 0 , where Tˆ = {Tk = {x¯ ∈ M into sets with the same first return time with respect to the first return map  f. The separation time is given by

s(x¯, y¯ ) := sup{k ≥ 0 :  f i (y¯ ) ∈ T (  f i (x¯ )), 0 ≤ i ≤ k},

. ∀x¯, y¯ ∈ M

For any x ∈ x¯ and y ∈ y¯ , it is reasonable to define s(x, y ) = s(x¯, y¯ ). It follows from hypothesis (3) that the map f is hyperbolic except for the indifferent fixed point p. Denote

λ := sup{|D fx |Exu |−1 o , |D f x |Exs |o : x ∈ M \ P }, |D fx |Exu |o = supv∈Exu ,v =0 |D|vfx|vr |r and |D f x v|r supv∈Exs ,v =0 |v| . It is evident that λ ∈ (0, 1). r where

P∗

{[P , P , . . . , P

159

(4.2)

|D fx |Exs |o =

We introduce a Banach space on M:

, L := { : supp ⊂ M

L := ∞ + ||s < ∞},

where  · L is the norm, ||s is a semi-norm given by

||s := sup

 x¯,y¯ ∈M

|(x¯ ) − (y¯ )| , λθ s(x¯,y¯ )

(4.3)

160

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

where m ≥ 2, α < m − 1 satisfying (4), and m − 1 − α ≤ θ ≤ 1. Remark 4.2. It is evident that L contains Hölder functions with . exponent θ and the support contained in M Lemma 4.1. Assume α < m − 1 satisfying (4), and m − 1 − α ≤ θ ≤ ¯ , f , P ) is irreducible and measure pre1. The Markov map (M, B , μ serving. There exists C > 0 such that for any  ∈ L and  ∈ L∞ with , one has supp ⊂ M

∞        ( k) ] Cor ( ,  ; f , μ ¯ ) − μ ¯ [ S    n k=n+1

≤ CFη (n ) ∞ L ,

(4.4)

 where S(k) is specified in (3.19), ∞ ¯ [S(k ) ] ≈ n−(η−1 ) , η = mα−1 , k=n+1 μ and Fη (n) is specified in Theorem 4.1. This lemma will be verified by applying Theorem 4.1. The proof is split into four parts. First, we show that  f has a useful property. Second, we study the Jacobian of  f with respect to the measures μ¯ . Third, we consider a property of the first return time function. Fourth, we estimate the measure of the sets with the same first return time. Proof. It follows from hypothesis (1) that the Markov map (M, B, μ¯ , f , P ) is irreducible. So the Markov map is irreducible measure preserving, since μ is an invariant measure for f. Step 1. We show that  f has big image property, that is, the measures of the images of the elements of the partition P 0 under  f are bounded away from 0. This is derived from the fact that the Markov partition is finite. Step 2. We verify that − log Jμ¯  (  f ) is locally θ -Hölder continuM ous (see also [1,44] for locally Hölder continuity), where J (  f) = μ¯ M

 ¯ . By the definition of the locally θ -Hölder continuous, dμ ¯M  ◦ f /d μ  M to show that a function is locally θ -Hölder continuous is equivalent to prove that this function is in L, which is introduced in (4.3). By applying similar arguments used in Lemma 2 in Section 3.1 in [49] (we combine the distortion estimate (3.18) and the method in [49]), we have that  f admits an absolutely continuous invariant  with the density function   on M measure μ h with respect to υ¯ , and the density function satisfies log  h ∈ L and is bounded away from 0 and infinity. By uniqueness of the invariant measure, which is derived from the assumption that f is topologically mixing, we   is equal to μ know that μ ¯M  on M. So, to study the locally Hölder continuity of μ ¯M  is equivalent to consider the same properties of . Direct calculation tells us that μ

    h◦ f, Jμ  ( f ) = Jν¯ ( f )  h

construction, we have that Z+ ⊂ {τ (x ) : x ∈ M \ P } (please refer to Fig. 2 for the illustration diagram with m = 2), implying that the greatest common divisor of {τ (x¯ ) − τ (y¯ ) : x¯, y¯ ∈ M} is one. Step 4. We prove that μ ¯ [S(k ) ] = O(1/kη ). Let γ u ∈ Wu (S) be any unstable manifold. Denote by μuγ the conditional measure of the SRB measure μ when it is restricted to γ u . By Proposition 3.3, the distortion of f along any unstable manifold is uniformly bounded. dμuγ Similar statements also hold for the density function . Hence, dνγu by (3.20), there exist C1 , C1 > 0 such that

C1 C1 ≤ μuγ (γku ) ≤ η , kη k where γku = γ u ∩ S(k ) , γ u is any unstable manifold in Wu (S), and S is specified in (3.19). By Fubini theorem and Proposition 3.2, one has that similar inequalities also hold for μ[S(k) ] with different positive constant coefficients, that is, there exist two positive constants B 1 and B1 such that

B 1 B1 ≤ μ[ S ( k ) ] ≤ η . kη k This gives that

∞

(4.5)

¯ [S k=n+1 μ

(k ) ] ≈ n−(η−1 ) .

Hence, the Markov map (M, B , μ ¯ , f , P ) is irreducible, topologically mixing, and probability preserving, the map  f has the big image property, log Jμ¯  (  f ) is locally θ -Hölder continuous, and M

μ¯ [S(k) ] = O(1/kη ) with η > 1. Therefore, the assumptions of Theorem 6.3 in [21] are satisfied, yielding that the statement of this lemma is correct by Theorem 6.3 in [21]. The proof is completed.  4.3. Polynomial decay rates for diffeomorphisms In this section, we shall establish the polynomial decay rates of correlation functions for the almost Anosov diffeomorphisms by using the results in previous sections. Now, we introduce a type of Hölder functions



Hθ : =  :

∃H > 0 s.t. |(x ) − (y )|

≤ H |x − y|θ and supp() ⊂ M \ P }, where m ≥ 2, α < m − 1 satisfying (4), and m − 1 − α < θ ≤ 1. Set

M0 := P = {P, P1 , · · · , Pl } and Mk :=

k 

f −i (M0 ),

(4.6)

i=0

where the part on the left hand side is the Jacobian of  f with re, and Jν¯ (  spect to μ f ) is the Jacobian of  f with respect to ν¯ . The following conclusion is obtained in [29]: Let h be a C 1+α∗ diffeomorphism on the smooth manifold and h preserve an SRB measure μ. Suppose that ξ is a partition subordinate to Wu . Then, for μ-a.e. x, the density ρ x of μξ with respect to ν x satisfies



∞ −1 u ρx ( y ) i=0 |Dhyi |Eyi |d = ∞ , ∀y, z ∈ ξ (x ), −1 u ρx ( z ) i=0 |Dhzi |Ez |d i

where yi = h−i (y ) and zi = h−i (z ), i ≥ 0, and ν is the Riemannian measure on M. So, log  h and log( h◦  f ) are in L. This, together with  the fact that log Jν¯ ( f ) is in L by (4.1) and Proposition 3.3, implies   that log Jμ ( f ) = − log Jμ  ( f ) is also in L, yielding that − log Jμ ( f ) ¯M  is locally θ -Hölder continuous. Step 3. We show that the greatest common divisor of {τ (x¯ ) − τ (y¯ ) : x¯, y¯ ∈ M} is one. By the properties of the map f and our

where k is any positive integer. For any n ∈ Z, let k = k(n ) be a number smaller than n, which will be given later. For any ,  ∈ Hθ , by direct calculation, one has

|Corn (,  ; f, μ )| = |Corn−k (,  ◦ f k ; f, μ )| ≤ |Corn−k (,  ◦ f k ; f, μ ) − Corn−k (,  k ; f, μ )| + |Corn−k (,  k ; f, μ ) − Corn−k (k ,  k ; f, μ )| + |Corn−k (k ,  k ; f, μ )| = : ( A1 ) + ( A2 ) + ( A3 ),

(4.7)

and

|Corn (,  ; f, μ )| ≥ (A3 ) − (A1 ) − (A2 ), where  k is a function defined by  k |A := inf{(x ) : x ∈ f k (A )} for any A ∈ M2k , k is defined analogously, and k := d (( f k )∗ (k μ )) , dμ

where k μ is a signed measure, whose density with

respect to μ is k .

X. Zhang / Chaos, Solitons and Fractals 123 (2019) 149–162

Finally, we show Theorem 2.2. The proof is divided into five steps. The first step studies the diameter of the set f k (M2k (x )). The second, third, and fourth step deal with the terms (A1 ), (A2 ), and (A3 ) in (4.7), respectively. The last step finishes the proof of Theorem 2.2. Step 1. We consider the diameter of the set f k (M2k (x )), which is useful in the estimate of (4.7). Since f is uniformly hyperbolic on k f −i (P )) gives an upper bound MP, the diameter of the set f k (∩2i=0 k of the diameter of the set f (M2k (x )). Now, we show 1

k −i diam( f k (∩2i=0 f (P ))) ≤ Cd k− α ,

k −i diam( f k (∩2i=0 f (P ))|W s ) ≤ Cs (κ s )k ,

where Cs is a positive constant. On the other hand, let us estimate k f −i (P )) along the unstable direction. The the diameter of f k (∩2i=0 upper bound for the diameter along the unstable direction is given by the length of the unstable manifolds contained in P with first return time to MP bigger than k. This, together with (3.20) and Proposition 3.2, yields that

k −i diam( f k (∩2i=0 f (P ))|W u ) ≤ Cu k− α , 1

where Cu is a positive constant. Hence, we obtain (4.8). Step 2. We study (A1 ). By direct calculation and (4.8), one has

|Corn−k (,  ◦ f k ; f, μ ) − Corn−k (,  k ; f, μ )|     ≤  ( ◦ f k −  k ) ◦ ( f n−k ) · dμ      +  ( ◦ f k −  k )dμ · dμ ≤ (2 max || )



|( ◦ f k −  k )|dμ ≤ (2 max || ) ·

θ

kα

. (4.9)

| − k |dμ = (2 max | | )|μ − k μ|m (M )

= (2 max | | )|( f k )∗ (( ◦ ( f k )μ ) − ( f k )∗ (k μ )|m (M ) ≤ (2 max | | )|( ◦ f k − k )μ|m (M )  = (2 max | | ) | ◦ f k − k |dμ ≤ (2 max | | ) ·

Cdθ θ

kα

.

(4.10)

Step 4. We investigate (A3 ) and show that

|Corn−k (k ,  k ; f, μ )| = |Corn−k (k ,  k ; f , μ¯ )|.

(4.11)

Since  and  are constant along stable manifolds contained in any Pi , we treat  and  as functions defined on M. This, together with f ◦ π = π ◦ f and π∗ (k μ ) = k (π∗ μ ), implies that



( k ◦ ( f n−k ))k dμ =

 

=  =

=  = and



k dμ



( k ◦ ( f n−k ))d (( f k )∗ (k μ ))  k d (( f n−k )∗ ( f k )∗ (k μ ))  k d (( f n )∗ (k μ ))

 k dμ =



 k d (π∗ ( f n )∗ (k μ )) n

 k d (( f )∗ (k μ¯ )) n

 k ◦ f · k dμ¯

d (( f k )∗ (k μ )) ·   = k dμ¯ ·  k dμ¯ .



 k dμ¯

This proves (4.11). Consequently, Corn−k (k ,  k ; f, μ ) can be determined by functions, which are constant along stable manifolds on each Pi ∈ P. Step 5. We prove (2.3). By Lemma 4.1 and (4.11), one has

A

(n − k )η−1

≤ (A3 ) = |Corn−k (k ,  k ; f , μ ¯ )| ≤

A

(n − k )η−1

, (4.12)

where A and A are two positive constants. Furthermore, since α < m − 1 satisfying (4), and m − 1 − α < θ ≤ 1, we have that αθ > mα−1 − 1 = η − 1. Therefore, taking k = [n/2], by (4.7), (4.9)–(4.12), and the fact Fη (n ) = O(1/nη ), we obtain

A(,  ) A (,  ) ≤ |Corn (,  ; f, μ )| ≤ , nη−1 nη−1

Acknowledgments

Cdθ

    Corn−k (,  k ; f, μ ) − Corn−k (k ,  k ; f, μ )          ≤  ( k ◦ ( f n−k ))( − k )dμ +   k dμ · ( − k )dμ 



where A (,  ) and A(,  ) are two positive constants determined by  and  . This verifies (2.3). This completes the whole proof of Theorem 2.2.

Step 3. We consider (A2 ). Denote by | · | the total variation of a signed measure. By using the fact (( f k )∗ (( ◦ ( f k )μ ) = μ and (4.8), one has

≤ (2 max | | )

 =

(4.8)

k f −i (P )) where Cd is a positive constant. The diameter of f k (∩2i=0 along the stable direction is obtained by using hypothesis (3), that is,

161

We would like to thank the reviewers for their thoughtful comments and efforts towards improving our manuscript. This research is partially supported by the National Natural Science Foundation of China (Grant 11701328), the Shandong Provincial Natural Science Foundation, China (Grant ZR2017QA006), and Young Scholars Program of Shandong University, Weihai (Grant 2017WHWLJH09). References [1] Aaronson J, Denker M, Urbanski M. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans Amer Math Soc 1993;337:495–548. [2] Alves J, Bonatti C, Viana M. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent Math 20 0 0;140:351–98. [3] Alves J, Dias C, Luzzatto S, Pinheiro V. SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Society 2017;19:2911–46. [4] Alves J, Leplaideur R. SRB measures for almost axiom a diffeomorphisms. Ergod Theory Dyn Sys 2016;36:2015–43. [5] Araújo V, Melbourne I, Varandas P. Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps. Commun Math Phys 2015;340:901–38. [6] Araújo V, Pacifico M, Pujals E, Viana M. Singular-hyperbolic attractors are chaotic. Trans Am Math Soc 2009;361:2431–85. [7] Baladi V. Positive transfer operators and decay of correlations. Adv Ser Nonlinear Dyn 20 0 0;16. World Scientific. [8] Baladi V, Liverani C. Exponential decay of correlations for piecewise cone hyperbolic contact flows. Commun Math Phys 2012;314:689–773. [9] Bálint P, Melbourne I. Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. J Stat Phys 2008;133:435–47. [10] Barreira L, Pesin Y. Lectures on Lyapunov exponents and smooth ergodic theory. University Lecture Series, 23; 2002. American Mathematical Society. [11] Benedicks M, Young L. Sinai-Bowen-Ruelle measure for certain Hénon maps. Invent Math 1993;112:541–76. [12] Benedicks M, Young L. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 20 0 0;261:13–56. [13] Bessa M, Rocha J, Varandas P. Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles. Dyn Syst 2018;33:691–707.

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