RECURRENCE RATE IN RAPIDLY MIXING DYNAMICAL SYSTEMS

BENOIT SAUSSOL

Abstract. For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.

1. Introduction

1.1. Decay of correlations. Let (X, f, μ) be a measure preserving dynamical system. Recall that the system is said to be mixing if for any functions φ, ψ in L^{2} the covariance

Cov(φ ∘ f^{n}, ψ) := ∫ φ ∘ f^{n}ψdμ − ∫ φdμ ∫ ψdμ → 0 as n → ∞.  (1)

The decay of the correlation function is, in great generality, arbitrarily slow. The notion of rapid mixing needs a little more structure.

Assume that X is a metric space with metric d, and consider the space Lip(X) of real Lipschitz functions on X. For many dynamical systems an upper bound for (1) of the form ∥φ∥∥ψ∥θ_{n} has been computed, where θ_{n} → 0 with some rate, and ∥ · ∥ is a norm on a space of functions with some regularity. Without loss of generality we are considering in this paper the rate of decay of correlations for Lipschitz observables^{1}.

A broad class of systems enjoy exponential decay of correlations. The main result of the paper (Theorem 3) applies to systems with super-polynomial decay of correlation. This includes for example Axiom A systems with equilibrium states, hyperbolic systems with singularities with their SBR measures such as those considered by Chernov in [7], many systems with a Young tower [17, 18], expanding maps with singularities such as in [14], some non-uniformly expanding maps [1], etc. The main reference for these questions is certainly the book by Baladi [2]. The reader will also find in the review by Luzzatto [12] an exposition of the recent methods for non-uniformly expanding systems and an extensive bibliography on this active field.

1.2. Recurrence rate and dimensions. The return time of a point x ∈ X under the map f in its r-neighborhood is

τ_{r}(x) = inf{n ≥ 1: d(f^{n}x, x) < r}.

We are interested in the behavior as r → 0 of the return time. We define the lower and upper recurrence rate as the limits

\underline{R}(x) = lim inf_{r→0} \frac{log τ_{r}(x)}{log(1/r)} and \overline{R}(x) = lim sup_{r→0} \frac{log τ_{r}(x)}{log(1/r)}.

1For example an immediate approximation argument allows easily to go from Hölder or class C^{k} to Lipschitz.

Whenever \underline{R}(x) = \overline{R}(x) we denote by R(x) the value of the limit.

From now on we assume that X is a Borel subset of a finite dimensional Euclidean space E. Denote by HD(Y) the Hausdorff dimension of a set Y ⊂ X. We define the Hausdorff dimension of a Borel probability measure μ by

HD(μ) = inf{HD(Y): Y Borel set s.t. μ(Y) = 1}

We also define a local version of the dimension, namely

\underline{d}_{μ}(x) = lim inf_{r→0} \frac{log μ(B(x, r))}{log r} and \overline{d}_{μ}(x) = lim sup_{r→0} \frac{log μ(B(x, r))}{log r}  (2)

It is well known that the Hausdorff dimension satisfies the relation

HD(μ) = ess-sup \underline{d}_{μ}.  (3)

Barreira and Saussol established in [4] the following relation

Proposition 1. Let f be a measurable map and μ be an invariant measure for f. The recurrence rates are bounded from above by the pointwise dimensions :

\underline{R} ≤ \underline{d}_{μ} and \overline{R} ≤ \overline{d}_{μ} μ-a.e.

We refer to the works by Boshernitzan [6] and Ornstein and Weiss [13] for pioneering related results.

In this paper we are giving conditions under which the opposite inequalities will hold, establishing the equalities

\underline{R} = \underline{d}_{μ} and \overline{R} = \overline{d}_{μ} μ-a.e.  (4)

1.3. Statement of the results.

Definition 2. We say that (X, f, μ) has super-polynomial decay of correlations if we have

|∫ φ ∘ f^{n}ψdμ − ∫ φdμ ∫ ψdμ| ≤ ∥φ∥∥ψ∥θ_{n}  (5)

with lim_{n} θ_{n}n^{p} = 0 for all p > 0, where ∥ · ∥ is the Lipschitz norm.

The main result of the paper is the following.

Theorem 3. Let (X, f, μ) be a measure preserving dynamical system. If the entropy h_{μ}(f) > 0, f is Lipschitz (or piecewise Lipschitz with some condition ; see Lemma 13) and the decay of correlation is super-polynomial then

\underline{R} = \underline{d}_{μ} and \overline{R} = \overline{d}_{μ} μ-a.e.

We postpone the proof at the end of Section 3. This extends some results by Barreira and Saussol in [4, 5], including the case of Axiom A systems with equilibrium states. The hypotheses in Theorem 3 are satisfied in a number of systems such as those already quoted in the introduction. All these systems have in common some hyperbolic behavior. We now give an example of a relatively different nature, due to the possibility of zero Lyapunov exponents, where one can still apply Theorem 3.

Example 4 (Ergodic toral automorphisms). Recall that any matrix A ∈ Sl(k, ℤ) (i.e. the entries of A are in ℤ and | det A| = 1) gives rise to an automorphism f of the torus 𝕋^{k} by f(x) = Ax mod ℤ^{k} which preserves the Lebesgue measure. The map f is ergodic if and only if the matrix A has no eigenvalue root of unity. Lind’s established [11] the exponential decay of correlations (using the algebraic nature and Fourier transform) which is more than enough to apply Theorem 3 and get

R(x) = k for Lebesgue a.e. x ∈ 𝕋^{k}.

for any ergodic automorphism of the torus, even non-hyperbolic.

Let f be a diffeomorphism of a compact manifold M and μ be an ergodic invariant measure. By Oseledec’s multiplicative ergodic Theorem the Lyapunov exponents

λ(x, v) = lim \frac{1}{n} log |d_{x}f^{n}v|

are well defined for μ-a.e. x ∈ M and all nonzero v ∈ T_{x}M and take a finite number of values called the Lyapunov exponents of the measure μ. Recall that a measure μ is said to be hyperbolic if none of its Lyapunov exponents are zero. Barreira, Pesin and Schmeling [3] prove the following.

Proposition 5. Let f be a diffeomorphism of a compact manifold and μ be an ergodic hyperbolic measure. Then we have

\underline{d}_{μ} = \overline{d}_{μ} = HD(μ) μ-a.e.

The case of an hyperbolic measure with zero entropy is completely understood.

Proposition 6. let f be a diffeomorphism of a compact manifold and μ be an hyperbolic invariant measure. If h_{μ}(f) = 0 then R = 0 = HD(μ) μ-a.e.

Proof. Barreira and Saussol established in [4] the inequality \overline{R} ≤ \overline{d}_{μ} μ-a.e. and it follows from Ledrappier and Young’s work [10] that HD(μ) = 0 if h_{μ}(f) = 0, which allows to conclude by Proposition 5. □

Corollary 7. Let f be a diffeomorphism of a compact manifold and μ be an hyperbolic measure with super-polynomial rate of decay of correlation. Then we have

R = HD(μ) μ-a.e.

Proof. If the entropy is zero then this is the content of Proposition 6. If the entropy is non-zero then this is the content of Theorem 3. □

We point out that in the case of interval maps with nonzero Lyapunov exponent, Saussol, Troubetzkoy and Vaienti prove that R = HD(μ) μ-a.e. for ergodic measures, under very weak regularity conditions [15]. See Remark 14-(i) for related results.

We now give a sketch of the strategy adopted in this paper.

Theorem 8 states that under sufficiently rapid mixing the recurrence rates equal the pointwise dimensions a.e. on the set where \underline{R} > 0. Indeed, mixing implies that μ(B ∩ f^{−n}B) = O(μ(B)^{2}) for large n. If now we consider the set B ∩ (f^{−n}B ∪ f^{−n−1}B ∪ ⋯ ∪ f^{−n−ℓ}B) then its measure is bounded by O(ℓμ(B)^{2}). If ℓ ≤ μ(B)^{−1+ε} then we get that the proportion of points inside B that enter in B in the time interval [n, n + ℓ] is bounded by O(μ(B)^{ε}). Using the decay of correlations we are able to prove that this last statement is true for n of the order diam(B)^{−δ} for some small δ > 0, whenever B is a ball. A Borel Cantelli argument then shows that typical points do not enter again in the ball B in the time interval [diam(B)^{−δ}, μ(B)^{−1+ε}] (see Lemma 9 for precise statement). This is what we call the long flight property.

In addition, for systems which are not too wild (e.g. finite Lyapunov exponents, see Lemma 13) and with nonzero metric entropy, a symbolic coding (see Lemma 12) allows to use Ornstein-Weiss’ theorem on repetition time of symbolic sequences to prove that the return time of a typical point in a ball B is larger than diam(B)^{−δ} ; see Lemma 11. This together with the long flight property establishes Equation (4).

The structure of the paper is as follows. We state and prove in Section 2 the core result, Theorem 8. In Section 3 we provide some conditions under which the recurrence rate is nonzero.

2. Rapid mixing implies long flights

Theorem 8. Assume that the rate of decay of correlations is super-polynomial. Then on the set \{\underline{R} > 0\} we have

\underline{R} = \underline{d}_{μ} and \overline{R} = \overline{d}_{μ} μ-a.e.

Proof. By Proposition 1 we know that \underline{R} ≤ \underline{d}_{μ} and \overline{R} ≤ \overline{d}_{μ}. Furthermore, the first inequality implies that for any a > 0 we have \{\underline{R} > a\} ⊂ \{\underline{d}_{μ} > a\} μ-a.e. But on the set \{\underline{R} > a\} we have τ_{r}(x) ≥ r^{−a} provided r is sufficiently small. By Lemma 9 below with δ = a and ε > 0 we get that τ_{r}(x) ≥ μ(B(x, r))^{−1+ε} provided r is sufficiently small, for μ-a.e. x ∈ \{\underline{R} > a\}. Thus \underline{R} ≥ (1 − ε)\underline{d}_{μ} and \overline{R} ≥ (1 − ε)\overline{d}_{μ} μ-a.e. on \{\underline{R} > a\}. The conclusion follows by taking ε > 0 arbitrary small. □

The following lemma expresses that the orbit of a typical point has the long flight property.

Lemma 9. Let X_{a} = \{\underline{d}_{μ} > a\} for some a > 0. For any δ, ε > 0, for μ-a.e. x ∈ X_{a} there exists r(x) > 0 such that for any r ∈ (0, r(x)) and any integer n in [r^{−δ}, μ(B(x, r))^{−1+ε}] we have d(f^{n}x, x) ≥ r.

Proof. Let D be the dimension of the Euclidean space E ⊃ X. Fix b > 0, c = aε/2 and consider for r_{0} > 0 the set G = G_{1} ∩ G_{2} ∩ G_{3} where

\begin{aligned}G_{1} & = \{x ∈ X_{a}: ∀r ≤ r_{0}, μ(B(x, 2r)) ≤ r^{a}\} \\ G_{2} & = \{x ∈ X: ∀r ≤ r_{0}, μ(B(x, r/2)) ≥ r^{D+b}\} \\ G_{3} & = \{x ∈ X: ∀r ≤ r_{0}, μ(B(x, r/2)) ≥ μ(B(x, 4r))r^{c}\}. \end{aligned}

We claim that μ(G) → μ(X_{a}) as r_{0} → 0. Indeed, by definition of the lower pointwise dimension we have μ(G_{1}) → μ(X_{a}). In addition since \overline{d}_{μ} ≤ D a.e. we have μ(G_{2}) → 1 and since E is Euclidean the measure μ is weakly diametrically regular (see Lemma 1 in [4]), thus μ(G_{3}) → 1 as well. Let r ≤ r_{0} and define the set

A_{ε}(r) = {y ∈ X: ∃n ∈ [r^{−δ}, μ(B(y, 3r))^{−1+ε}], d(f^{n}y, y) < r}.

Let x ∈ G. By the triangle inequality we get the inclusion

\begin{aligned}B(x, r) ∩ A_{ε}(r) & ⊂ \{y ∈ B(x, r): ∃n ∈ [r^{−δ}, μ(B(x, 2r))^{−1+ε}], d(f^{n}y, x) < 2r\} \\ & = ⋃_{r^{−δ}≤n≤μ(B(x,2r))^{−1+ε}} B(x, r) ∩ f^{−n}B(x, 2r). \end{aligned}

Let η_{r}: [0, ∞) → ℝ be the r^{−1}-Lipschitz map such that 1_{[0,r]} ≤ η_{r} ≤ 1_{[0,2r]} and set φ_{x,r}(y) = η_{r}(d(x, y)). Clearly φ_{x,r} is also r^{−1}-Lipschitz. By the assumption on the decay of correlation we obtain

μ(B(x, r) ∩ f^{−n}B(x, 2r)) ≤ ∫ φ_{x,r}φ_{x,2r} ∘ f^{n}dμ
≤ ∥φ_{x,r}∥∥φ_{x,2r}∥θ_{n} + ∫ φ_{x,r}dμ ∫ φ_{x,2r}dμ
≤ r^{−2}θ_{n} + μ(B(x, 2r))μ(B(x, 4r)).

Choose p > 1 such that δ(p − 1) − 2 ≥ D + 2b and take r_{0} so small that n ≥ r_{0}^{−δ} implies θ_{n} ≤ (p − 1)(n + 1)^{−p}. Since ∑_{n≥q} n^{−p} ≤ \frac{1}{p−1}(q − 1)^{1−p} we obtain for r ∈ (0, r_{0})

\begin{aligned}μ(B(x, r) ∩ A_{ε}(r)) & ≤ r^{δ(p−1)−2} + μ(B(x, 2r))^{ε}μ(B(x, 4r)) \\ & ≤ μ(B(x, r/2)) (r^{b} + r^{εa−c}) . \end{aligned}

Let B ⊂ G be a maximal r-separated set^{2}. Since (B(x, r))_{x∈B} covers G we have

μ(G ∩ A_{ε}(r)) ≤ ∑_{x∈B} μ(B(x, r) ∩ A_{ε}(r))

≤ ∑_{x∈B} μ(B(x, r/2))(r^{b} + r^{εa−c})

≤ r^{b} + r^{εa−c}

since the balls (B(x, r/2))_{x∈B} are disjoints. This implies that

∑_{m} μ(G ∩ A_{ε}(e^{−m})) < ∞,

thus by Borel-Cantelli Lemma we obtain that for μ-a.e. y ∈ G there exists m(y) such that for every m ≥ m(y) we have y ∉ A_{ε}(e^{−m}). For any r sufficiently small (i.e. r ≤ e^{−m(y)}), taking m such that e^{−m−1} < r ≤ e^{−m} implies that e^{δm} ≤ r^{−δ} and 3e^{−m} < 3er hence there exists no n ∈ [r^{−δ}, μ(B(y, 3er))^{−1+ε}] such that d(f^{n}y, y) < r. By weak diametric regularity the factor 3e in the radius is irrelevant and this proves the lemma. □

Remark 10. Observe that we only use that the decay of correlation is at least n^{−p} for some p > \frac{D+2}{δ} + 1. If in addition (5) holds with the first norm ∥φ∥ taken to be the L^{1}(μ) norm (e.g. expanding maps) then p > \frac{D+1}{δ} + 1 suffices.

2that is if x ≠ x′ ∈ B then d(x, x′) ≥ r and maximal in the sense that for any y ∈ G there exists x ∈ B such that d(x, y) < r.

3. Non-zero recurrence rate

We proceed now to find conditions under which the recurrence rate does not vanish. Denote by ξ(x) the unique element of a partition ξ containing the point x and by ξ^{n} = ξ∨f^{−1}ξ∨⋯∨f^{−n+1}ξ the dynamical partition, for any integer n .

3.1. Coding by symbolic systems. Next lemma, which proof is fairly simple, is the keyobservation which gives to Theorem 8 all its interest.

Lemma 11. Assume that ξ is a partition such that [Large interior property] for μ-a.e. x there exists χ = χ(x) < ∞ such that B(x, e^{−χn}) ⊂ ξ^{n}(x) for all n sufficiently large.

If furthermore the entropy h_{μ}(f, ξ) > 0 then \underline{R} > 0 μ-a.e.

Proof. Let ξ be such a partition. Define

R_{n}(x, ξ) = min{k > 0: f^{k}x ∈ ξ^{n}(x)}.

Ornstein and Weiss [13] prove that if ξ is a finite partition with entropy h_{μ}(f, ξ) then

lim_{n→∞} \frac{1}{n} log R_{n}(x, ξ) = h_{μ}(f, ξ) μ-a.e.

Since ξ has large interior, for μ-a.e. x ∈ X there exists a number χ = χ(x) such that B(x, e^{−χn}) ⊂ ξ^{n}(x). Thus^{3}

\underline{R}(x) = lim inf_{n→∞} \frac{log τ_{e^{−nχ(x)}}(x)}{nχ(x)} ≥ lim inf_{n→∞} \frac{log R_{n}(x, ξ)}{nχ(x)} = \frac{h_{μ}(f, ξ)}{χ(x)} > 0 μ-a.e.

□

Combining Lemma 11 and Theorem 8 we get that if we have super-polynomial decay of correlations and a partition of positive entropy with large interior then \underline{R} = \underline{d}_{μ} and \overline{R} = \overline{d}_{μ}. This includes for example the case of loosely Markov dynamical systems and we recover Urbanski’s result in [16]^{4}. The rest of the section consists in finding sufficient conditions for the existence of such a partition.

3.2. Reasonable dependence on initial condition.

Lemma 12. Assume that the system (X, f, μ) satisfies the following condition [Reasonable sensitivity] for μ-a.e. x there exists γ, λ > 0 such that f^{n} is e^{λn}- Lipschitz on the ball B(x, e^{−γn}) for all n sufficiently large.

If furthermore the entropy h_{μ}(f) > 0 then there exists a partition which satisfies the conditions in Lemma 11 (i.e. large interior and nonzero entropy).

3Note that by monotonicity of τ_{r} the liminf may be attained by any subsequence of the form e^{−χn}.

4Strictly speaking, these systems may only enjoy a local super-polynomial decay of correlations, in the sense that there exists a partition (modulo μ) into open sets V_{i} and constants θ_{n}^{i} such that (5) holds whenever supp φ ⊂ V_{i} and supp ψ ⊂ V_{i}, where lim_{n} θ_{n}^{i}n^{p} = 0 for all p > 0. It is clear from the proof of Lemma 9 that this local property suffices.

Proof. Claim : For any x ∈ X, s > 0 there exists ρ ∈ (s, 2s) such that for all n

μ(\{y ∈ X: ρ − 4^{−n}s < d(x, y) < ρ + 4^{−n}s\}) ≤ \frac{1}{2^{n−1}} μ(B(x, 2s)).  (6)

Indeed, let m be the measure on the interval (0, 2) defined by m([0, t)) = μ(B(x, st)). We construct a sequence of open intervals I_{n} starting from I_{0} = (1, 2). If I_{n} is an interval of length 4^{−n} we divide it into 4 pieces of equal length and choose I_{n+1} the left of the right central piece of smallest measure. We have m(I_{n+1}) ≤ \frac{1}{2}m(I_{n}). I_{n} is a decreasing sequence of intervals with \overline{I}_{n+1} ⊂ I_{n} thus ∩_{n}I_{n} contains one point, say ρ̄. Since ρ̄ ∈ I_{n} we have ρ̄ ± 4^{−n} ∈ I_{n−1} thus m((ρ̄ − 4^{−n}, ρ̄ + 4^{−n})) ≤ m(I_{n−1}) ≤ \frac{1}{2^{n−1}} m(I_{0}). Proving the claim with ρ = sρ̄.

Fix s > 0 so small that any partition made by sets of diameter less than 2s has nonzero entropy (see [8]). Choose a maximal s-separated set E. For any x ∈ E take ρ_{x} ∈ (s, 2s) such that (6) in the claim holds. Let E = {x_{1}, x_{2}, …} be an enumeration of the (at most) countable set E. Put B_{i} = B(x_{i}, ρ_{x_{i}}) and define Q_{1} = B_{1}, Q_{2} = B_{2} \ Q_{1}, Q_{3} = B_{3} \ (Q_{1} ∪ Q_{2}), … By maximality the collection of sets ξ = {Q_{1}, Q_{2}, …} is a partition of X (modulo μ) and since ∂ξ ⊂ ∪_{i}∂B_{i} we get

\begin{aligned}μ(\{x ∈ X: d(x, ∂ξ) < 4^{−n}s\}) & ≤ μ(∪_{i}\{x ∈ X: ρ_{x_{i}} − 4^{−n} < d(x_{i}, x) < ρ_{x_{i}} + 4^{−n}\}) \\ & ≤ \frac{1}{2^{n−1}} ∑_{i} μ(B(x_{i}, 2s)). \end{aligned}

Since the x_{i} are s-separated and E is Euclidean there are at most c(E) = c(dim E) balls of radius 2s that can intersect, thus the last sum is bounded by \frac{c(E)}{2^{n−1}} . This proves that for some constants a, c > 0 and all ε > 0

μ(x ∈ X: d(x, ∂ξ) < ε) < cε^{a}.

Thus for any b > 0 we have by the invariance of μ

∑_{n} μ({x ∈ X: d(f^{n}x, ∂ξ) < e^{−bn}}) ≤ ∑_{n} ce^{−abn} < ∞.

This implies by Borel-Cantelli Lemma that for μ-a.e. x there exists n(x) < ∞ such that d(f^{n}x, ∂ξ) ≥ e^{−bn}, hence B(f^{n}x, e^{−bn}) ⊂ ξ(f^{n}x), for any n ≥ n(x). Taking c(x) ∈ (0, 1) sufficiently small we have B(f^{n}x, c(x)e^{−bn}) ⊂ ξ(f^{n}x) for all integer n.

Fix x ∈ X where the reasonable sensitivity condition holds. Without loss of generality, and changing if necessary c(x) into a smaller constant we assume that f^{n} is e^{λn}-Lipschitz on the ball B(x, c(x)e^{−γ}n) for all integer n and that λ > γ + b.

We show then by induction that B(x, c(x)e^{−λn}) ⊂ ξ^{k}(x) for any k ≤ n. Indeed, this is trivially true for k = 1, and if this holds for some k ≤ n − 1 then we have

f^{k}(B(x, c(x)^{2}e^{−γn}) ⊂ B(f^{k}x, c(x)e^{λk−γn}) ⊂ B(f^{k}x, e^{−bn}) ⊂ ξ(f^{k}x).

Hence B(x, c(x)^{2}e^{−γn}) ⊂ ξ^{k+1}(x). □

We finally provide a sufficient condition for reasonable sensitivity.

Lemma 13. Assume that f is Lipschitz or that the following condition holds [Piecewise Lipschitz with finite exponent] there exists a partition 𝒜 (modulo μ) into open sets such that on each A ∈ 𝒜 the map f is Lipschitz with constant L_{f}(A), the singularity set ∂𝒜 = ∪_{A∈𝒜} ∂A is such that μ({x ∈ X: d(x, ∂𝒜) < ϵ}) ≤ cϵ^{a} for some constants c > 0 and a > 0 and the average Lipschitz exponent log L_{f} := ∑_{A∈𝒜} log^{+} L_{f}(A)μ(A) is finite.

Then the first condition of Lemma 12 is satisfied (i.e. the system is reasonably sensitive).

Proof. We prove the piecewise case, the other one is obvious. Let λ > log L_{f}. By the Birkhoff Ergodic Theorem, for μ-a.e. x there exists m(x) such that

L_{f}(𝒜(x))L_{f}(𝒜(fx)) ⋯ L_{f}(𝒜(f^{n−1}x)) ≤ e^{λn}

for all n ≥ m(x). Replacing if necessary the upper bound by e^{λn}/c(x) for some constant c(x) ≥ 1 the inequality will hold for any integer n. Proceeding as in the last part of the proof of Lemma 12 we get that for any b > 0, changing c(x) if necessary, we have B(f^{n}x, c(x)e^{−bn}) ⊂ 𝒜(f^{n}x) for any integer n. We then conclude similarly that B(x, c(x)^{2}e^{−bn}e^{−λn}) ⊂ 𝒜^{n}(x). This concludes the proof taking γ = b + λ. □

The proof of Theorem 3 follows now easily from the preceding results.

Proof of Theorem 3. By Lemma 13 the map is reasonably sensitive. This implies by Lemma 12 the existence of a partition with large interior. By Lemma 11 we find that \underline{R} > 0 a.e. and the conclusion follows from Theorem 8. □

Remark 14. (i) We remark that if f is C^{2} on a compact manifold, or more generally if f is piecewise C^{1+α} with reasonable singularity set such as in [9], and μ is an ergodic measure with nonzero entropy, then the exponents λ and γ in Lemma 12 can be taken arbitrarily close to the largest Lyapunov exponent λ_{μ}^{+} of the measure^{5}. Thus the exponent χ in Lemma 11 may also be taken arbitrarily close to λ_{μ}^{+}. This readily implies that

\underline{R} ≥ h_{μ}/λ_{μ}^{+} μ-a.e.  (7)

This is optimal in dimension one or more generally for conformal maps, where under mild assumptions we have HD(μ) = h_{μ}/λ_{μ}.

In the case of diffeomorphisms a similar argument may be applied also with backward iterates and the lower bound in (7) easily generalizes^{6} to

\underline{R} ≥ h_{μ} (\frac{1}{λ_{μ}^{+}} − \frac{1}{λ_{μ}^{−}} ) μ-a.e.

where λ_{μ}^{−} is the smallest Lyapunov exponent of the measure μ. Note that for C^{2} surface diffeomorphisms (e.g. Henon maps) with hyperbolic measures this is optimal and shows that

R = HD(μ) μ-a.e.

5to see this, consider a Lyapunov chart whose local chart at x has a diameter ρ(x), where ρ is η-slowly varying.

A choice like λ = λ_{μ}^{+} + 2η and γ = λ + η would do the job.

6it essentially amounts to consider a two-sided version of Lemma 11 ; we leave the details to the reader.

(iii) Combining the above observation with Remark 10 shows that the assumption on the superpolynomial decay of correlations in Theorem 8 may be reduced to a decay at a rate n^{−p} for some p > \frac{D+2}{h_{μ}}λ_{μ}^{+} + 1.

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E-mail address: benoit.saussol@u-picardie.fr

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