Recurrence rate in rapidly mixing dynamical systems

. 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.


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 [16,17], expanding maps with singularities such as in [13], 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 [11] 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 recurrence rate as the limits and R(x) = lim sup r→0 log τ r (x) log(1/r) .
Date: November 2004. 1 For example an immediate approximation argument allows easily to go from Holder or class C k to Lipschitz. 1 Whenever R(x) = R(x) we denote by R(x) the value of the limit. From now on we assume that X is a finite dimensional Euclidean space. Denote by HD(Y ) the Hausdorff dimension of a set Y ⊂ X. We define the Hausdorff dimension of a probability measure µ by HD(µ) = inf{HD(Y ) : µ(Y ) = 1} We also define a local version of the dimension, namely It is well known that the Hausdorff dimension satisfies the relation 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 : We refer to the works by Boshernitzan [6] and Ornstein and Weiss [12] for pioneering related results.
In this paper we are giving conditions under which the opposite inequalities will hold, establishing the equalities R = d µ and R = d µ µ-a.e.
1.3. Statement of the results.
Definition 2. We say that (X, f, µ) has super-polynomial decay of correlations if we have with lim n θ n /n p = 0 for all p > 0, where · is the Lipschitz norm. We say that the local decay of correlations is super-polynomial if there exists a partition (modulo µ) into open sets V i and sequences θ i n such that (5) holds whenever supp ϕ ⊂ V i and supp ψ ⊂ V i , where lim n θ i n /n p = 0 for all p > 0. 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 finite average Lipschitz exponent ; see Definition 15) and the (local) decay of correlation is super-polynomial then 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 theorem also applies to loosely Markov dynamical systems and we recover Urbanski's result in [15]. 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 [10] the exponential decay of correlations (using the algebraic nature and Fourier transform) which is more than enough to apply Theorem 3 and get for any ergodic automorphism of the torus, even non-hyperbolic.
Let f be a diffeomorphism of a compact manifold M and µ be an invariant measure. By Oseledec's multiplicative ergodic Theorem the Lyapunov exponents 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 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 invari- Proof. Barreira and Saussol established in [4] the inequality R ≤ d µ µ-a.e. and it follows from Ledrappier and Young's work [9] 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 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 [14]. See Remark 17-(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 R > 0. Indeed, mixing implies that then we get that the proportion of points inside B that never enter in B in the time interval [n, n + ℓ] is bounded by 2µ(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. This is what we call the long fly property. A Borel Cantelli argument then shows that typical points do have long flies (see Lemma 9 for precise statement). If in addition we also have R > δ then it immediately shows that the return time into small neighborhoods B cannot be much less (at an exponential scale) than µ(B) −1 , establishing Equation (4).
On the other hand, for systems which are not too wild (e.g. finite Lyapunov exponents, see Lemma 16) and with nonzero metric entropy, a symbolic coding (see Lemma 14) 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 not less than diam(B) −δ ; see Lemma 12.
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.

Rapid mixing implies long flies
Theorem 8. Assume that the local rate of decay of correlations is super-polynomial. Then on the set {R > 0} we have Proof. By Proposition 1 we know that R ≤ d µ and R ≤ d µ . Furthermore, the first inequality implies that {R > a} ⊂ {d µ > a} µ-a.e. But on the set {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 ∈ {R > a}. Thus R ≥ (1 − ε)d µ and R ≥ (1 − ε)d µ µ-a.e. on {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 fly property. Lemma 9. Let X a = {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. For clarity we assume that the (global) rate of decay of correlation is super-polynomial. The obvious modifications in the proof would consits essentially in considering separately each sets G ∩ {x ∈ V i : d(x, ∂V i ) > ν} for arbitrarily small ν > 0 in place of the unique set G defined below.
Let x ∈ G. By the triangle inequality we get the inclusions 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 Choose p > 1 such that δ(p − 1) − 2 ≥ D + 2b and take r 0 so small that n ≥ r −δ Let B ⊂ G be a maximal r-separated set 2 . Since (B(x, r)) x∈B covers G we have since by the balls (B(x, r/2)) x∈B are disjoints. This implies that m µ(A ε (e −m )) < ∞, thus by Borel-Cantelli Lemma we obtain that for µ-a.e. y ∈ G there exists m(y) such for every m > m(y) there exists no n ∈ [e −δm , µ(B(y, 3e −m )) −1+ε ] such that d(f n y, y) < e −m . By weak diametric regularity (and changing slightly if necessary the values of ε and δ), this proves the lemma.
Remark 10. Observe that we only use that the decay of correlation is at least n −p for some p > 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 > D+1 δ + 1 suffices.

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 : partitions with large interior. Definition 11. We say that a partition ξ has large interior if for µ-a.e. x there exists χ = χ(x) < ∞ such that B(x, e −χn ) ⊂ ξ n (x) for all n sufficiently large.
Next lemma, which proof is fairly simple, is the key-observation which gives to Theorem 8 all its interest.
Lemma 12. If there exists a partition with large interior and nonzero entropy then R > 0 µ-a.e.
Proof. Let ξ be such a partition. Define Ornstein and Weiss [12] prove that if ξ is a finite partition with entropy h µ (f, ξ) then 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 Combining Lemma 12 and Theorem 8 we get that if we have local super-polynomial decay of correlations and a partition of positive entropy with large interior then R = d µ and R = d µ . The rest of the section consists in finding sufficient conditions for the existence of such a partition.

3.2.
Reasonable dependence on initial condition. Definition 13. We say that a system (X, f, µ) is reasonably sensitive if 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. Lemma 14. If the system (X, f, µ) is reasonably sensitive and the entropy h µ (f ) > 0 then there exists a partition with large interior and nonzero entropy.
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 ) ≤ 1 2 m(I n ). I n is a decreasing sequence of intervals with I n+1 ⊂ I n thus ∩ n I n contains one point, sayρ. Sinceρ ∈ I n we haveρ ± 4 −n ∈ I n−1 thus m((ρ − 4 −n ,ρ + 4 −n )) ≤ m(I n−1 ) ≤ 1 2 n−1 m(I 0 ). Proving the claim with ρ = sρ.
Fix s > 0 so small that any partition made by sets of diameter less than 2s has nonzero entropy. 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 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 2s)).
Since the x i are s-separated and X is Euclidean there are at most c(X) = c(dim X) balls of radius 2s that can intersect, thus the last sum is bounded by c(X) 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 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 sensitovoty 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 Hence B(x, c(x) 2 e −γn ) ⊂ ξ k+1 (x).
We finally provide a sufficient condition for reasonable sensitivity. 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(f x)) · · · 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 14 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 16 the map is reasonably sensitive. This implies by Lemma 14 the existence of a partition with large interior. By Lemma 12 we find that R > 0 a.e. and the conclusion follows from Theorem 8.

Remark 17. (i)
We remark that if f is C 1 on a compact manifold, or more generally if f is piecewise C 1+α with reasonable singularity set such as in [8], then the exponents λ and γ in Definition 13 can be taken arbitrarily close to the largest Lyapunov exponent 3 λ + µ . Thus the exponent χ in Lemma 12 may also be taken arbitrarily close to λ + µ . This readily implies that R ≥ h µ /λ + µ . This is optimal in dimension one or more generally for conformal maps, where under mild assumptions we have HD(µ) = h µ /λ µ .
(ii) 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 > D+2 hµ λ + µ + 1.