Every expanding measure has the nonuniform specification property

Exploring abundance and non lacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the lenght of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfy the nonuniform specification property. As consequences, we (re)-obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents and we prove that any expanding measure is limit of Dirac measures on periodic points.


Introduction
A basic problem in Dynamical Systems and its Applications is to study the existence and abundance of periodic points and understand its distribution on the underlying space. In simple terms, for a differentiable map f in a Riemman manifold M, ones want to analyse which conditions on f and its derivatives ensure the existence of periodic points and how they are distributed in M.
For uniformly hyperbolic maps, Bowen stablished in [3] that the asymptotical exponential growth of the set P n ( f ) of periodic points of period n was determined by the topological entropy: He introduced the important notion of specification by periodic orbits and proved a number of important results concerning the uniqueness and the ergodic properties of equilibrium states, asymptotic growth and the limit distribution of periodic orbits and so on, for Axiom A diffeomorphisms and flows ( [3]). In words, a system has the specification property if a (small) error is fixed, given a piece of orbit of size n there exists a periodic points that follows (up to this error) this orbit up to the moment n and has period n + K, where K depends only on the error and do not depend on n.
More precisely, we say that f has the specification property if there exists ǫ 0 > 0 such that for all x ∈ M and 0 < ǫ < ǫ 0 , n ≤ 1, there exists a periodic point p ∈ M such • d( f i (p), f i (x)) < ǫ, for i = 0, . . . , n • p has period less than n + K, with K = K(ǫ) depending only on ǫ.
For convenience, we define the dynamical ball of size ǫ and lenght n by and this can be rephrased just saying that there exists a periodic point of period n + K(ǫ) in B n (x, ǫ).

INTRODUCTION
Beyond the uniformly hyperbolic setting, the understanding of periodic orbits and its structure is much less developed. A remarkable work stablishing connections between Lyapunov exponents of a given measure and periodic points and its distribution was obtained by Katok in [6]. There, he proved a technical lemma known as Katok Closing Lemma: roughly, it tell us that if f is C 1+α diffeomorphism and x is a recurrent point in a Pesin's block such that f n (x) is in the same Pesin's block, the orbit of x up to the moment n can be shadowed by a periodic point p with period depending only on the shadowing constant and the Pesin block, not on n (see [6], Section 3 for precise statements). Recent improvements of this result include the papers [14,5].
Here, we improve part of this results relaxing the hypothesis in [6] and obtaining a quatitative version of Katok's Closing Lemma for non-uniformly expanding maps preserving a ergodic measure with positive exponents. We study the extension of the Bowen's specification property in a measure-theoretical setting.
Consider an ergodic invariant measure µ with only positive Lyapunov exponents for any C 1 endomorphisms f with non-flat critical set. We are able to show that for µ almost every point, given a natural number n and ǫ > 0 there exists a periodic point p on the dynamical ball B n (x, ǫ) with period K(n, ǫ) that growth assymptotic like n at infinity.
This generalize the Katok´s Closing Lemma in several ways. First, we are able to deal with C 1 maps, instead C 1+α maps, since we do not need to make use of Pesin´s theory. Moreover, we are able not only to prove the existence of a shadowing point but to obtain quantitative estimates on its period. As consequence of our result, we are able to obtain estimates on Poincaré recurrence in terms of the Lyapunov exponents. Let us describe it in detail.
The study of recurrence and return times are among the most prolific tools for better understanding of statistical properties of dynamical system. The most basic concept in this context is the Poincaré recurrence of a set. Given a measurable dynamical system (M, µ, f ) and a measurable set A ⊂ M, we define its Poincaré recurrence as In the literature, many relations have been established between recurrence times and other important aspects of dynamical systems such as entropy, Hausdorff dimension, mixing properties and Lyapunov exponents.
In order to grasp a finer understanding of these relations it is useful consider return times associated to shrinking neighborhoods such as decreasing sequences of balls or cylinders.
Closely related to the Poincaré recurrence of B n (x, ǫ) we may ask for the existence of periodic points in this dynamical ball. Sometimes it is possible to find a periodic point z in a given ball but often its period is unrelated to n or ǫ. In the scenario we are facing to, we can restrict our attention to investigate how frequently dynamical balls contain periodic points of (at some extent) controlled period. This gives place to a notion of specification that bounces back to Bowen itself [3,4].
In [13] it is presented the nonuniform counterpart of these notions allowing that the radius ǫ in Equation (1) decrease with n. To begin with, let q : M → [0, ∞) be a η-slowly varying function, that is to say, a function satisfying q( f (x)) ≤ e η q(x), for all x ∈ M and some fixed η > 0. A (n, ǫ, q) nonuniform dynamical ball is defined as And we say that ( f , µ) has the nonuniform specification property if for µ-almost everywhere, given a ηslowing varying function q, the ball B n (x, ǫ) contains a periodic point whose period is less than n + K(n, ǫ, η) satisfying lim η→0 lim sup n→∞ K(n, ǫ, η) n = 0.
In the same work Saussol et al [13] proved that the nonuniform specification property implies an estimate of the recurrence time for arbitrary positive µ-measure sets in terms of Lyapunov exponents.
Here we are able to obtain a more natural result showing that positiveness of all Lyapunov exponents implies the nonuniform specification property. The main idea that we use to obtain this result, is the 2 PRELIMINARIES notion of hyperbolic time. This notion has been used by many authors to obtain statistical properties of dynamical systems, such as existence and uniqueness of SRB measures ( [2]), stochastic stability ([1]), infinite Markov Partitions ( [11]) and equilibrium states ( [7,8]). Along this paper, we prove that almost every point with respect to any ergodic measure with positive exponents has a nonlacunary sequence of hyperbolic times. Let us state our main result: Theorem A. Let f : M → M be a C 1 map with non-flat critical set C. Any f -invariant expanding measure µ satisfies the nonuniform specification property.
See Section 2 for precise definitions. Using Theorem A above and the Theorem of [13], we obtain that: Moreover, if we assume that h µ ( f ) > 0 the following inequality also holds true It is interesting to observe that the inequalities presented in Corollary 1.1 may be attained, but there are examples where these inequalities are strict. In fact, there exists a linear expanding map on the two dimensional torus preserving the Lebesgue measure and with Lyapunov exponents 0 < λ 1 < λ 2 such that 1 for Lebesgue almost every point x ∈ M. See Section 4 of [13], for further details and some other examples. Another interesting consequence of Theorem A above, is the fact that every expanding measure is approximated in the weak ⋆ topology by Dirac measures at periodic points. We point out that in [15] the author make use of the notion of nonuniformly specification to establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, he obtained upper and lower bounds for the measure of deviation sets of some nonuniformly expanding maps, in-cluding quadratic maps and robust multidimensional nonuniformly expanding local diffeomorphisms. To describe more precisely the notions in this introduction, we recall some basic notions in Ergodic Theory in the next section.

Preliminaries
By a classical theorem due to Oseledets ([9]), given a f -invariant measure µ, for almost every point x ∈ M there exists an f -invariant (measurable) splitting Note that the last expression implies that each λ i is an f -invariant function, i.e., λ i (x) = λ i ( f (x)). In particular, if µ is ergodic the functions λ i are constant almost everywhere. Let be the numbers λ j (x), in a non-decreasing order, and each repeated with multiplicity dim(E i (x)). These numbers are called the Lyapunov exponents of f at the point x. We recall that the support of a invariant measure µ is the full measure set supp(µ) of all points such any neighbourhood has positive measure.

Positive Exponents and Hyperbolic Times
Throughout we assume that f : M → M be a C 1 map with non-flat critical set C which preserves an expanding ergodic invariant measure µ. We also assume that f is strongly transitive on the support of µ. It is well-known (see [10]) that expanding measures admit invariant unstable local manifolds at almost every point. However, the dependence of these manifolds with the base point is just measurable and several of its features are not suitable for computations. In particular, the size of this local manifold is just a measurable function and this is an additional challenge when we need to handle these objects. In the next definition, we introduce a concept that address some of these difficulties: Definition 3.1. Given c > 0 and δ > 0, we say that n is a (c, δ)-hyperbolic time for a point x ∈ M, if we can find a neighborhood V n (x) of x such that f n : V n (x) → B δ ( f n (x)) is a homeomorphism with the property that given x 1 , In this context, we call V n (x) = f −n B δ ( f n (x)) the hyperbolic pre-ball of length n and radius δ at x.  3.3. Given 0 < θ < 1, let H θ (c, δ, f ) be the set of points with frequency of (c, δ)-hyperbolic times at least θ: From [2], we have a sufficient criterium for abundance of (c, δ)-hyperbolic times at a point x. We say that f is asymptotically c-expanding at a point x, if lim sup Furthermore, we say that the point x satisfies the condition of slow approximation to the critical set if for all ǫ > 0 there exists δ > 0 such that Here, dist δ is the δ-truncated distance: dist δ (x, C) := dist(x, C), whenever dist(x, C) < δ and 1 otherwise .
Abundance of hyperbolic times is given by Proof. We observe that since the Lyapunov exponent of µ is positive, we may find a constant c > 0 such that for almost every x ∈ M, there exists n 0 (x) ∈ N such that if n ≥ n 0 (x) then D f n (x)v ≥ e 8cn v , for every v ∈ T x M. This is equivalent to D f n (x) −1 ≤ e −8cn . Denote by A k the set It is clear that µ(A c k ) goes to zero when k goes to infinity. Observe that On the other hand, Since log D f (x) −1 is integrable, by Birkhoff´s Ergodic Theorem, the function h k (x) = 1 converges in L 1 (dµ) to some function ϕ. Using that µ(A c k ) goes to zero when k goes to infinity, we have that lim Observing this and Equation (3) above, there exists l 0 such that if ℓ ≥ l 0 In particular, using Birkhoff´s Ergodic Theorem once more, the Condition (2) holds for the function f ℓ at almost every point x ∈ M. Since µ is an ergodic measure and has finite Lyapunov exponents with respect to µ, we have by Lemma 10.2 of [11] that Condition (3) holds for f ℓ at almost every point. Since Condition 2 and 3 are satisfied, we have by Lemma 3.4 that there exists θ > 0 such that almost every x ∈ M belongs to H θ (c, δ, f ℓ ).
We fix ℓ, c, δ, θ > 0 as in Lemma above, and consider g = f ℓ . When we say just hyperbolic time, we mean (c, δ)-hyperbolic time with respect to g. Define the first hyperbolic time function n 1 : H θ (c, δ, g) → N setting n 1 (x) as the first hyperbolic time of x. Remark 3.6. Observe that if m is a hyperbolic time for x and n is a hyperbolic time for f m (x), then n + m is a hyperbolic time for x. From this follows that if n 1 (x) < n 2 (x) < . . . denotes the sequence of hyperbolic times of x, then For the next result we make use of [11,Lemma 4.7]: Lemma 3.7. [11,Lemma 4.7] Let (G j ) j∈N be a collection of subsets of M such that for all x ∈ G n and 0 ≤ j < n we have that g j (x) ∈ G n−j . Let B ⊂ M and let x ∈ B be a point such that #{j ≥ 1; x ∈ G j and g j (x) ∈ B} = +∞.
Consider O + (x) the positive orbit of x and let T : where ϕ(y) = min{j ∈ N; y ∈ G j and g j (y) ∈ B}. Then, if lim sup n 1 n #{1 ≤ j ≤ n; x ∈ G j and g j (x) ∈ B} > θ > 0 Using this lemma we are able to prove that Lemma 3.8. The first hyperbolic time function of g is integrable: Proof. Observe that by Lemma 3.5, the set H θ (c, δ, g) has full measure. Since µ is ergodic, by Birkhoff's Ergodic Theorem, to show the integrability of n 1 is enough to verify that for every x ∈ H θ (c, δ, g) we have where we set n 0 (x) = 0. Indeed, taking G j = H j (c, δ, g) and B = H θ (c, δ, g) as in Lemma 3.7 above, it follows from Remark 3.6 that if x ∈ G n then g j (x) ∈ G n−j for every 0 ≤ j < n. Observe that by the definition, for every x ∈ H θ (c, δ, g) we have that lim sup n 1 n #{1 ≤ j ≤ n; x ∈ G j and g j (x) ∈ B} > θ > 0.
An increasing sequence (a k ) k∈N of natural numbers is called nonlacunary, if lim k→∞ a k+1 a k = 1.
In [8], this notion is used in the context of equilibrium states to prove existence and uniqueness of a special type of weak Gibbs measure, called nonlacunary Gibbs measure. Therein, they proved that the integrability of the first hyperbolic time implies nonlacunarity. Here, we slightly generalize this result. Let γ : R + → R + a bijection. We say that a increasing sequence (a k ) k∈N is γ-nonlacunary, if lim k→∞ a k+1 − a k γ(a k ) = 0.
In particular, if γ is the identity, a γ-nonlacunary sequence is just a nonlacunary sequence.
Before prove this Lemma, we remark that if we know a priori that µ H n (c, δ, f ) decays in particular way, then we may take γ(t) growing less than t at the infinity. For example, µ H n (c, δ, f ) decays expoentially, the hypothesis of the lemma above is easily satisfied for any function of the form γ(t) = t p , where p > 0.
Proof. Let D be the set of points for which the sequence n j (·) fails to be γ-nonlacunary. For each r > 0, define L r (n) = {x ∈ M : n 1 (x) ≥ rγ(n)}. If x ∈ D then there exists a rational number r > 0, and there are infinitely many values of i such that n i+1 (x) − n i (x) ≥ rγ(n i (x)). By Remark 3.6, the latter implies that So, there are arbitrarily large values of n such that x ∈ f −n (L r (n)). In other words, D is contained in the set ∞ k=0 n≥k f −n (L r (n)).
Since µ is invariant, we have µ( f −n (L r (n))) = µ(L r (n)) for all n. Then Thus, using the hypothesis that γ −1 (n 1 (·)/r) is integrable, By the Borel-Cantelli lemma, this implies that L has measure zero. It follows that µ(D) = µ(L) = 0, as claimed. Proof. Let us consider as before g = f ℓ with ℓ chosen in such a way that µ-almost all x ∈ M has a nonlacunary infinite sequence (n k ) k of g-hyperbolic times (Lemma 3.5 and Corollary 3.10). We can assume that x is such a point on supp(µ), the support of µ. Let ǫ > 0 and n ≥ 1 be fixed with ǫ < δ and let k be such that ℓn k < n ≤ ℓn k+1 . From the uniform continuity of f it follows that there exists γ = γ(ǫ) < ǫ such that for all y ∈ M and 0 ≤ k ≤ ℓ, As a consequence we have Let V(x) = g −n k+1 B(g n k+1 (x), γ) be the g-hyperbolic pre-ball around x of length n k+1 . Since n k+1 is a g-hyperbolic time for x and g −j is a contraction on B(g n k+1 (x), γ) for all 1 ≤ j ≤ n k+1 , it follows that g n k+1 (V(x)) = B(g n k+1 (x), γ) and that V(x) = B g,n k+1 (x, γ).
As f is strongly transitive on the support of µ, we can find r > 0 and N(ǫ) ∈ N such that for all y ∈ supp(µ) we have B(y, r) ⊂ f j (B(x, γ)) for some j ≤ N(ǫ).
This finish the proof of this lemma.

Proof of Theorem A
In this section we prove Theorem A. We need to prove that given an ergodic expanding measure µ, ǫ > 0, η > 0 small enough and q a η-slowly varying function, for µ-almost every point x ∈ M we have that the nonuniform dynamical ball B n (x, ǫ, q) has a periodic point with period less than n + K(n, x, ǫ, η), where lim η→0 lim sup n→∞ K(n, x, ǫ, η) n = 0.
Proof of Theorem A. By Lemma 3.5 and Corollary 3.10, there exists ℓ such that almost every x ∈ supp(µ) has infinitely many hyperbolic times n 1 < n 2 < . . . for g = f ℓ and that (n k ) is nonlacunary sequence. Assume that η < c and ǫ < δ. Denote by B n (x, ǫ, q) the nonuniform (n, ǫ, q)-dynamical ball with respect to f and by B ℓ n (x, ǫ, q) the (n, ǫ, q)-dynamical ball with respect to f ℓ . Observe that since D f uniformly is bounded from above and, by the Mean Value inequality, given ǫ > 0 we may choose α = α(ǫ) < ǫ such that given y ∈ M and 0 ≤ k ≤ ℓ, f k B(y, rα) ⊂ B( f k (y), rα), for every r > 0. As a consequence, given m ∈ N, we have that B ℓm (x, α(ǫ), q) ⊂ B ℓ m (x, ǫ, q).
Given n ∈ N big enough, take ℓn i ≥ n > ℓn i−1 two consecutive hyperbolic times for x. By the definition of η-slowly varying function, we have that q(g n (x)) ≤ e ℓnη q(x). Observing that g −j x is a contraction for 1 ≤ j ≤ n i , we have that