Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures

We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.


Introduction
In this paper, we build horseshoes for C 1+α mappings (not necessarily invertible) preserving ergodic hyperbolic measures with positive measure-theoretical entropy and then prove that the exponential growth rate of the number of periodic points is greater than or equal to the measure-theoretical entropy. This research is a natural generalization of Katok's argument in his paper [2]. We also prove that the exponential growth rate of the number of hyperbolic periodic points with "large" Lyapunov exponents is equal to hyperbolic entropy. Hyperbolic entropy means the entropy resulting from hyperbolic measures.
Horseshoes are exhibited as examples of systems that demonstrate complicated dynamical behaviors and allow us to model the behavior by a shift map over a finite alphabet. Thus, it is an interesting problem to consider the existence of horseshoes. Let M be a compact manifold of dimension 2 and f : M → M be a C 1+α diffeomorphism with positive entropy. Katok's argument illustrates the fact that positive entropy implies the existence of horseshoes and the entropy of these inner horseshoes can approximate h µ (f ), which means the underlying horseshoes demonstrate nearly the same complicated property as the whole systems. One might expect Katok's argument to be true for endomorphsims with all Lyapunov exponents not zero.
Gelfert [16] proved the existence of horseshoe for mappings with only positive Lyapunov exponents under some integrability conditions that are used to control the effect of critical points and singular points. We give a generalization from the nonsingular case to all Lyapunov exponents not zero, without integrability assumption on critical points (Theorem 1.3). Besides, we also control the Lyapunov exponents of periodic points in the horseshoe. After completing this paper, we came upon a paper by Y.M. Chung [34] who dealt with the same problem, but the starting point of our proof is shadowing lemma for sequences maps which is different from the idea used in the proof of [34]. Also, [34] does not give results on controlling the Lyapunov exponents of periodic points which is used in the proof of Theorem 1.6 in this paper. Now let us state our main results. Let M be closed d−dimensional Riemannian manifold.
Definition 1.1. For any continuous map T on metric space N , the inverse limit space N T of (N, T ) is the subset of N Z consisting of all full orbits, i.e.
There exists a natural metric defined as Thus N T is a metric space with norm satisfing max i d(x i , y i ) ≥ d(x,ỹ) ≥ d(x 0 , y 0 ). LetT be the shift mapT ((x i ) i∈Z ) = (x i+1 ) i∈Z on N T . From Lemma 2.8, the set of invariant measures of (T , N T ) and the set of invariant measures of (T, N ) are equivalent. Denoteμ as the extension measure for µ. This extension also keeps entropy, i.e. hμ(T ) = h µ (T ). (2) ifλ 1 >λ 2 > · · · >λ k are the distinct Lyapunov exponents of µ, with multiplicities n 1 , · · · , n k ≥ 1, denoteλ the same as before, then there exists a dominated splitting on TxM = ⊔T π(f nx ) M,x ∈H where ⊔ means the disjoint union, TxM = E u ⊕ E s , and there exists N ≥ 1 such that for each i = 1, 2 eachx ∈H and each unit vector v ∈ E u (π(x)), u ∈ E s (π(x)), where P n (f ) denotes the number of periodic points with period n.
Theorem 1.6 below is a generalization of a result by Chung and Hirayama [35]. They proved that the topological entropy of a C 1+α surface diffeomorphism is given by the growth rate of the number of periodic points of saddle type. We prove here that for any C 1+α mappings on any dimensional manifold, the growth rate of the number of hyperbolic periodic points equals to the entropy coming from hyperbolic measures, hyperbolic entropy (see Defnition 1.5).
We point out that there is a similar result concerning the topological pressure for diffeomorphisms in Gelfert and Wolf's paper [18]. They proved that, for C 1+α diffeomorphisms, topological pressure for potentials with only hyperbolic equilibrium states is totally determined by the value of potentials on saddle periodic points with "large" Lyapunov exponents. where P H(n, f, K, a) means the number of collection of periodic points with n period and with uniform (K, a)-hyperbolicity (see Definition 3.12).
Now we give a short discussion about the main techniques we used in this paper. As stated before, the starting point of our proof of Theorem 1.3 is the shadowing lemma for sequences of mappings. In Avila, Crovisier and Wilkinson's new paper [4], they give a compact proof of shadowing lemma using the shadowing lemma for sequences of mappings and then they establish a direct way to find a horseshoe by coding some special separated set directly. Inspired by their ideas, we establish a shadowing property for extension mapf in the inverse limit space M f which inherits many properties for the mapping f and then construct horseshoes in the inverse limit space.
Finally, we end this section with a short note about critical points. The key issue caused by critical points is the switch of the unstable direction and the stable direction which will cancel the hyperbolicity. Such occurs, for example, in snapback repellers. Such phenomenon highly conflicts with the existence of absolutely continuous invariant measures (acim). Thus there are many compositions, which concern the existence of acim, taking critical points into account carefully ( [20,13,27] ). Nevertheless, by Mañe's multiplicative ergodic theorem, the derivatives along the unstable directions of almost all orbits in a Pesin block are isomorphisms. So, in terms of the shadowing lemma and the construction of horseshoes, the only collapse may happen here is along the stable direction which does not affect the shadowing lemma.

Acknowledgement
I would like to thank Amie Wilkinson for fruitful conversations. She is not only a research mentor but also a spiritual mentor to me. I would also like to thank Zhihong Xia for useful conversations in the preparation of this paper. This work was done during my stay in The University of Chicago. I would like to thank The University of Chicago for their hospitality and China Scholarship Council for their financial support.
2. Preliminaries 2.1. Inverse limit space. First we give the definition of regular Anosov mappings to illustrate some differences between diffeomorphisms and mappings in dynamical features.
Definition 2.1. [10]A regular map f ∈ C 1 (M, M ) is an Anosov mapping if there exist constants C > 0, 0 < λ < 1 and a Riemannnian metric < ·, · > on T M such that for every f -orbit ( xn that is preserved by the derivative Df and satisfies the conditions: Remark 2.2. It is noticeable that we do not ask for a splitting of the whole tangent bundle T M = E s ⊕ E u . It may happen that E u (xn) n∈Z = E u (yn) n∈Z though x 0 = y 0 . There is a construction of a mapping that is close to an algebraic Anosov mapping while at a point it has many different local unstable manifolds in [10]. This construction can also be reckoned as an explanation of the non stability of Anosov mappings. For E s , E s (xn) n∈Z only depends on x 0 . Of course, there are special systems for which E u x does not depend on the orbits containing x. A classical example of such mapping is any algebraic mapping of the torus, such as n 1 1 1 for The following theorem is the classical Oseledec's theorem, a version of the Multiplicative Ergodic Theorem for differentiable mappings. (1) There is a measurable integer function r : G → Z + with r • f = r.
(2) For any x ∈ G, there are real numbers Remark 2.4. By Oseledec's ergodic theorem 2.3 for maps, we only have a filtration type splitting in the tangent space which kills lots of skills in Pesin theory. Thus, there only exist well defined stable manifolds for mappings. Nevertheless, it is comforting that by Pugh and Shub's theorem 2.9 below, we can find full measure orbits such that along these orbits, there exist well defined invariant unstable manifolds. It is worth to note that what underlies this fact is the multiplicative ergodic theorem for non-invertible maps given by Ruelle or Mañe's [9,31].
It is a common idea to consider the inverse limit space for mappings. Letf : Thus, there is a natural measurable cocycle over f associated with Df . We abuse notation Df : M f × Z → L(d, R) defined as following.
Definition 2.6. The measurable cocycle Df over f is defined as following Remark 2.7. We should notice that inverse limit space isn't a manifold. It is just a topological space with linear cocycleDf over it and the dimension of M f is even infinite usually. Although we can not sayDf is the derivative tof , it is a linear cocycle overf .
Invariant measures in M f can be projected down to invariant measures in M by projection π. The following lemma says M inv M f is equivalent to M inv M .
, satisfying the following properties: The splitting is measurable with respect tox and the angles between any two associated subspaces vary sub-exponentially under iteration, i.e.
Although we do not use it in this paper, we state a result in [22] about the existence of unstable manifolds along orbits. Let f : M → M be a C 1+β mapping and let µ be an invariant measure for f with no zero exponent, then for almost all full orbits of f there are stable and unstable disc families which are Borel, vary sub-exponentially along orbits and are invariant. Remark 2.10. It is worth to note that the understanding of dynamics in inverse limit spaces is far away from the understanding of the original maps. For example, it is well known that a non-invertible mapping on a compact manifold is in general not stable except it is expanding [10]. Even so, for Anosov mappings, the dynamical structure of its orbit space is stable under C 1 small perturbations [29][32].

2.2.
Shadowing lemma for sequences of mappings. A lot of shadowing problems can be reduced to the following "abstract" shadowing problem. Let H k be a sequence of Banach spaces (k ∈ Z or k ∈ Z + ), we denote by | · | norms in H k and by || · || the corresponding operator norms for linear operators. Let us emphasize that the spaces H k are not assumed to be isomorphic.
Consider a sequence of mappings It is assumed that the values |φ k (0)| are uniformly small, say, |φ k (0)| ≤ d. We are looking for a sequence v k ∈ H k such that φ k (v k ) = v k+1 and the values |v k | are uniformly small, for example, the inequalities sup k |v k | ≤ Ld hold with a constant L independent of d.

Construction of Horseshoe
In this section, we focus our attention on the setΛ which is given in Proposition 2.9.
Definition 3.1. Let µ be an ergodic hyperbolic probability measure for a C 1 mapping f : M → M . Correspondingly, we have inverse limit space M f , shift mapf and ergodic measureμ with respect to µ. A compact positive measure setΛ(η) ⊂Λ is called a η-uniformity block for µ(with tolerance η > 0) if there exists K > 0 and a measurable map C η :Λ → GL(d, R) which is continuous on subsetΛ(η) such that: , for eachx ∈Λ and n ∈ Z.

Lemma 3.4. [3]
(Tempering-Kernel Lemma) Let f : X → X be a measurable transformation. If K : X → R is a positive measurable tempered function, then for any ǫ > 0, there exists a positive measurable function K ǫ : X → R such that K(x) ≤ K ǫ (x) and , · · · , λ r(x) (x) ∈ R and l 1 (x), · · · , l r(x) (x) ∈ N depending only onx with l i (x) = d such that for every ǫ > 0 there exists a tempered map Proof. This result follows directly from the same proof as the diffeomorphism case (See Theorem S.2.10 [3]). We give a sketch here. Ifx ∈Λ then is the expanding part and E 2 (x 0 ) is the contracting part. Define a new scalar product on each E 1 (x 0 ) and ǫ > 0 as follows: where < ·, · > denotes the standard scalar product on R n ; If u, v ∈ E 2 (x 0 ) then where < ·, · > denotes the standard scalar product on R n . Now according to the definition of Lyapunov exponentsλ i (x), for eachx ∈Λ and ǫ > 0 there exists a constant C(x, ǫ) such that Similarly, we have To extend the scalar product to T x0 M , consider Applying v = C ǫ (x)u in inequality (2,3), we get It remains to prove that C ǫ (x) is tempered. Since the angles between the different subspaces satisfy a sub exponential lower estimate due to Theorem2.9, it is enough to consider just block matrices. Set B N := {x ∈Λ|||C ± ǫ (x)|| < N }. For some N > 0 large enough, by the Poincare Recurrence Theorem, there exists a set Y ⊂ B N such that µ(B N \Y ) = 0 and the orbit ofỹ ∈ Y returns infinitely many times to Y . Thus let m k be a sequence such thatf m k (x) ∈ Y for all k. Then and therefore for almost every pointỹ ∈ Y the spectra of A ǫ and Df are the same. Since N is chosen arbitrarily this is true for almost everyx ∈Λ. Observe that , so by taking the growth rates in both equations we find that lim n→∞ 1 n log ||C ± ǫ (f nx )|| = 0 for allx ∈Λ for which A ǫ and Df have the same spectrum.
Theorem 3.6. If f : M → M is C 1 mapping and µ is an ergodic hyperbolic probability measure for f , then for every η > 0 there exists a uniformity blockΛ(η) of tolerance η for µ. Moreover,Λ(η) can be chosen to have measure arbitrarily close to 1 with suitable choose of K.
Proof. Applying Oseledec and Pesin's Reduction theorem given in Theorem 3.5 and Lusin theorem, we can get this theorem.
The assumption of the existence of hyperbolic ergodic measure ensures hyperbolicity. The norm we used in the proof of Theorem 3.5 is called Lyapunov norm. In non-uniformly hyperbolic case, Lyapunov metric is a powerful technique. Under Lyapunov norm, one can get uniformly hyperbolic property. From the definition of uniformity blocks, we can see that for anyx ∈Λ(η) there exist linearization and diagonalizaton of f along orbitx. In the following theorem, we want to estimate the C 1 distance between diagonalization and our maps under local charts. Next theorem says for uniformity block there exists uniformly bound which depends only on the tolerance of the block. We need a promotion from points to their neighborhoods, which requires higher regularity C 1+α . Theorem 3.7. (Pesin's argument for mappings) Let f : M → M be a C 1+α mapping preserving the ergodic hyperbolic probability measure µ. LetΛ(η) be a uniformity block of tolerance η for µ . Then there exist K > 0, ξ 0 > 0, a measurable map C η :Λ → GL(d, R) which is continuous onΛ(η) and a measurable function ξ :Λ → R + which is also continuous onΛ(η) such that: (1) max{||C −1 η (f n (x))||, ||C η (f n (x))||} < K exp(η|n|), for eachx ∈Λ and n ∈ Z.
Hence if ||w|| is sufficiently small the construction of the nonlinear part of fx is negligible. In particular,
From the definition of δ η (x) we have Applying the Tempering-Kernel Lemma 3.4 to δ η (x) we find a measurable K η : is obviously an embedding. Condition (a) follows from the continuous property of C η onΛ(η) and condition (b) follows from the definition. We only need to prove (c) and (d). Actually, on the set φx(B(x 0 , K −1 ξ η (x))) , we have Now we give the proof of condition (d). From the definition of C η , we only need to proof the second inequality. For allx ∈Λ(η) and for any y, y ′ ∈ B(x n , K −1 ξ η (f n+1x )), we have Actually, the last inequality can be improved to the form stated in our theorem. This is a direct result from the continuity of C η .
The set B(x n , K −1 ξ(f nx )) are called Lyapunov neighborhoods of the orbitx at x n . Although it may happen that the restriction of f on Lyapunov neighborhoods may not be invertible, the degeneration happens only along the stable direction which does not affect shadowing mechanism. The size of Lyapunov neighborhoods decay slowly (at rate at most e −η ) for points along orbitx.
Proof. Our goal here is to construct sequences of mappings on space R d satisfying conditions in Lemma 2.12. For any (x n ) n∈Z , an ε pseudo orbit forf with jumps iñ Λ(η), denote x n = π(x n ). 0 < ε < ε 0 is determined later. Following the notation in the proof of Theorem 3.7, we have sequences of mappingsg n : . From Theorem 3.7, we have d(g n (v), L n (v)) ≤ η, ∀n ∈ Z, and any v ∈ φx(B(x n , K −1 ξ(x n ))). Denote Considering the jumping points, i.e. 0 < d(fx n ,x n+1 ) ≤ ε, by the continuity of φx onΛ(η) and φ ∈ C ∞ , we have It is easy to see that sequences of mappings Φ n also satisfy other conditions in Lemma 2.12, thus there is an unique sequence of points z n ∈ R d satisfying Φ n (z n ) = (z n+1 ). Then, {φ −1 xn z n } n∈Z is a real orbit under map f .
Asf : M f → M f is invertible and measure entropy comes from any positive measure subset, invariant or not, we have the following type of Katok's argument which says there exist horseshoes in inverse limit space M f . In order to clarify to process of projecting the horseshoe in the inverse limit space to the initial space, we give the construction here briefly. Our proof follows the the idea used in [4]. These ideas of constructing pseudo orbits also appeared in [7,14,15]. ( (2) ifλ 1 >λ 2 > · · · >λ k are the distinct Lyapunov exponents of µ, with multiplicities n 1 , · · · , n k ≥ 1, denoteλ the same as before, then there exists a dominated splitting on TxM = ⊔T π(f nx ) M,x ∈H: and there exists N ≥ 1 such that for each i = 1, 2 eachx ∈H and each unit vector v ∈ E u (π(x)), u ∈ E s (π(x)), Proof.
Step 1: For tha sake of estimating the distance of measures, we first give a finite collection of continuous functions onΛ that can be used in weak * metric. Let C(Λ) be the space of real valued continuous functions defined onΛ. Choose γ ∈ (0, δ) and letφ 1 ,φ 2 , · · · ,φ k be a finite collection of C(Λ) such thatṼ contains the set of probability measuresν satisfying: Step 2: By Theorem 1.1 in [2], measure-theoretic entropy is the exponential growth rate of the minimal number of Bowen ball covering a positive measure set. More specifically, given x ∈Λ, ρ > 0, n ∈ N, the (n, ρ) Bowen ball is defined as For any positive number ξ < 1, measure entropy (which is independent of ξ) is defined as hμ(f ) = lim sup n→∞ 1 n log N (n, ρ, ξ).
We also define (n, ρ)-separated set here. S(n, ρ) is called a (n, ρ)-separated set for set K if for any pointx ∈ K ⊂Λ, there exists a pointỹ ∈ S(n, ρ) such that d(f i (x),f j (x)) ≥ ρ, for some i ∈ [0, n − 1]. We will take advantage of the fact that the maximal number of (n, ρ)-separated set is bigger than the minimal number of (n, ρ) Bowen balls covering the same set.
Step 3: Now we are ready to give the scale of Bowen balls and the scale of separation, i.e. ρ in the definition of the Bowen ball and the separated set. Assume 0 < ε 1 < min( γ 2hμf +4 , δ 4 ). Choose ρ > 0 small enough and N 0 ∈ N such that for any n ≥ N 0 N (n, ρ,μ(Λ(η))/2) > e n(hμ(f)−ε1) and such that for any d(x, y) ≤ ρ, ∀x, y ∈Λ, The small number ρ here is a separation scale.
Step 5: We chose (n, ρ)-separated set coveringΛ(η) in this step. We need not only separation property but also a common return time toΛ(η), so that we can control the segments. We also use some combinatory techniques to estimate the number of (n, ρ)-separated segments with the common return time toΛ(η).
For each n ∈ N, let S(n, ρ) be a maximal (n, ρ)-separated set inΛ(η) n . Without of loss of generality, we can assume that each two points in S(n, ρ) come from different orbits (if there are two points in the same orbit, just give a small perturb of it). Then,Λ (η) n ⊂ ∪x ∈S(n,ρ) B(x, n, 2ρ) and for N 1 large enough such that for any n ≥ N 1 we get and let N ∈ [N 1 , (1 + ε)N 1 ] be the value of n maximizing ♯V n . Assuming N 1 large enough, Step 6: Now we have had lots of separated segments with common return time. But in order to construct pseudo orbits from the segments of these separated points, we need to chose separated segments coming in to and getting away from the same ball.
Step 7: Now we can construct pseudo orbits. Consider the set of all orbits whose segments of length N originate inỸ and end in B. Concatenating these strings defines a two sided shift σ l based on l symbols, which has topological entropy log l ≥ N (h(μ,f ) − 2ξ) − log t. We will construct a horseshoeH ⊂ M f such that f N |H has σ l as a topological factor. Considering the setỸ of all ε 2 −pseudo orbits of the form: whereỹ ij =ỹ ij+1 ∈Ỹ. Note that these are also ε 2 −pseudo orbits with jumps iñ Λ(η), sincef N (ỹ) ∈Λ(η), for allỹ ∈Ỹ. Each element ofỸ can be naturally encoded as an element of {1, · · · , l} Z × {0, · · · , N − 1}. We defineH to be the set ofx ∈ M f whosef −orbit Cε 2 −shadowing some pseudo orbits inỸ.
Hyperbolicity ofH follows from the fact that the orbit of anyx ∈H stays in the union of finitely many regular neighborhoods, on which f stays close to a uniformly hyperbolic sequence in {A ǫ (x j1 ), A ǫ (x j2 ), · · · , A ǫ (x jN )}. Other conclusions follows from our construction directly. Thus we finish the proof.
Corollary 3.11. Let f : M → M be a C 1+α mapping preserving an ergodic hyperbolic probability measure µ. We have lim sup where P n (f ) denotes the number of periodic points with period n.
and ||Df j f i (p)(v) || ≥ Ke ja ||v||, ∀v ∈ E u f i (p) . Let P H(n, f, K, a) be the collection of periodic points with period n and uniform (K, a)-hyperbolicity. Proof. The first part is a direct corollary. We only need to prove the second equality.
As periodic points for f can also be viewed as periodic points forf , we use the same notation. Assume the Lyapunov splitting over the orbit of periodic point p with period P (p) is Let I(p) be the index of p, i.e. I(p) is the dimension of stable bundle E s . We define the following collection of periodic points with uniform hyperbolicity and the same index, P H(n, f, a, K, I) = {p ∈ P n (f )|||Df i f j (p)) v|| ≥ Ke ia ||v||, ∀v ∈ E u f j (p) , ||Df i f j (p)) u|| ≤ K −1 e −ia ||u||, ∀u ∈ E s f j (p) , ∀0 ≤ j ≤ n − 1, I(p) = I}.
As the splitting on periodic points can be continuously extended to the closure set,f |(∪ n P H(n, f, a, K, I)) is uniformly hyperbolic. Thusf |∪ n P H(n, f, a, K, I) is an expansive (from unique shadowing property) homoeomorphism and then f |π(∪ n P H(n, f, a, K, I)) is an expansive map. From the fact that π(P H(n, f, a, K, I)) is a n-separated set, one has lim n→+∞ 1 n log ♯P H(n, f, a, K, I) ≤ h(f |π(∪ n P H(n, f, a, K, I)))).
From the principle of variation, for any ε > 0, there exists a hyperbolic measure µ supported on π(∪ n P H(n, f, a, K, I)) such that h(f |π(∪ n P H(n, f, a, K, I))) ≤ h µ (f ) + ε ≤ H(f ) + 2ε. From the arbitrary choice of ε, we obtain lim sup

Relation between exponential growth rate of periodic points and degree
Asymptotic growth rate of the complexity of the orbit structure attracts people's attention for a long time. There are several points of view to describe asymptotic behaviors of dynamical systems, such as topology, measure, homology, etc. Commonly, one cares about the growth rate of the number of periodic points, measure-theoretic entropy, topological entropy and the spectral radii of the action on homology, etc.
Let M be a compact connected d−dimensional manifold. For C 1 mapping f : M → M , Misiurewicz and Pryztycki [26] proved that h top (f ) ≥ log |deg(f )|.
Let P n (f ) denotes the number of periodic points with period n. Katok [2] proved that for any C ∞ surface diffeomorphism f : M → M we have lim sup n→+∞ 1 n log P n (f ) ≥ h top (f ).
Inspired by these two results, Shub posed an interesting case in the problem 3 of [24]. Let f be a smooth degree two C 1+α map on 2-shpere S 2 where α > 0. Problem (Shub) : Does lim sup n→+∞ 1 n log P n (f ) ≥ log 2 hold? In order to get periodic points, a usual technique is the closing lemma used by Katok in [2] which is based on the hyperbolicity of invariant measures. If we assume there exists a hyperbolic invariant ergodic measure µ of C 1+α map f with h µ (f ) ≥ log deg(f ), then from Corollary 3.11 we get lim sup n→+∞ 1 n log P n (f ) ≥ log deg(f ). There is also a direct corollary from Corollory 3.13 as follows. We give some notes for the case when M is a surface. It is easy to see that if f is a diffeomorphism, then every invariant measure with positive entropy is hyperbolic. But this might not be true for endmomorphisms. For noninvertible mappings on surface, one might can not get hyperbolic invariant measures with measure-theoretic entropy approximating topological entropy. It may usually happen that for noninvertible mapping f , such as examples given by Pugh and Shub in their new paper [6]. In other words, from the equality in Theorem 3.13, it is highly possible that the growth rate of the number of saddle periodic points is strictly smaller than topological entropy. But it is still possible that the growth rate of the number of periodic points with zero Lyapunov exponents is greater than degree. One may need some topological techniques to get Shub's question.