Statistics of return times: a general framework and new

applications

Masaki Hirata^{∗} Benoît Saussol^{†} Sandro Vaienti^{†}

July 1998

Abstract

In this paper we provide general estimates for the errors between the distribution of the first, and more generally, the Kth return time (suitably rescaled) and the Poisson law for measurable dynamical systems. In the case that the system exhibits strong mixing properties, these bounds are explicitly expressed in terms of the speed of mixing. Using these approximations, the Poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval.

1 Introduction

The investigation of asymptotically rare events is growing up as a new direction in the understanding of statistical properties of dynamical systems. By “asymptotically rare” events we mean, in a wide sense and following the terminology in the review paper of [Coe97], those events which have asymptotically zero probability but which occur with a well determined asymptotic limit law. In the dynamical setting, where we have a probability space (X, B, μ) with a measurable μ-preserving mapping T acting on it, the “events” will usually be the visits into a sequence of sets Ω_{k} ∈ B of positive measure but with their measure going to zero in the limit of large k. We call the event “rare”, when the expected entrance time in Ω_{k} diverges with k. A well-known result in ergodic theory shows how abundant are the “asymptotically rare” events. Let us consider in fact an ergodic measure μ for an endomorphism T and take a measurable subset Ω: then Kac’s theorem [CFS82] says that the expectation of the return time to Ω, starting from Ω, is just μ(Ω)^{−1}.

∗Mathematical Department,  Tokyo Metropolitan University,  Japan.

mhirata@comp.metro-u.ac.jp

†Centre de Physique Théorique, Luminy, Marseille and PHYMAT, Mathematical Department, University of Toulon, France. saussol@cpt.univ-mrs.fr, vaienti@cpt.univ-mrs.fr

Kac’s theorem suggests the good normalization to keep in order to study the asymptotic distribution of the return time to Ω. The natural object will thus be the distribution:

F_{Ω}(t) = μ_{Ω} (x ∈ Ω | τ_{Ω}(x)μ(Ω) > t)  (1)

where τ_{Ω}(x) is the first return time to Ω provided that x ∈ Ω and μ_{Ω} is the normalized restriction of μ to Ω. The question will be whether the limit of F_{Ω}(t) exists when the measure goes to zero and what kind of distribution is recovered. The condition that the starting point x in (1) belongs to Ω could be relaxed by asking that x belongs to the whole space. In this case, F_{Ω}(t) will give the distribution of the “visiting time” into Ω, but in order to get its asymptotic distribution, a suitable normalization is needed [GS97]. The situations sketched above could be considerably refined, producing richer processes (see the quoted paper [Coe97] for an historical account of these questions and an exhaustive bibliography). We will however explore some of them in this paper under a more general perspective and successively by giving applications to class of systems never investigated before.

Let us first come back to formula (1) and replace Ω with a decreasing sequence of neighborhoods of a given point z ∈ X, Ω_{ε}(z), such that their measure goes to zero when ε → 0^{+}. Then for some classes of hyperbolic dynamical systems, notably axiom A diffeomorphisms [Hir93], transitive Markov chains [Pit91], expanding maps of the interval with a spectral gap [Col96] and in the more general setting of systems verifying a strong mixing property (“self-mixing” condition and φ-mixing [Hir95]), and recently even in the case of rational maps with critical points in the Julia set [Hay98a], it is possible to prove that the distribution F_{Ω_{ε}(z)}(t) goes to the exponential-one law e^{−t} and this for μ-almost every z ∈ X. A strong improvement of this kind of result appears in the paper [GS97], where an upper bound for the difference

|μ (τ_{A}(x) > \frac{t}{μ(A)λ(A)}) − e^{−t}|

was explicitly computed in the case of φ-mixing systems and where A is a cylinder set, and λ(A) a suitable normalizing factor. Recently [Hay98b] obtained an exponential error estimate for the quantity like (1) in the case of parabolic rational maps.

To enrich the process, and the statistics, one successively introduce the Kth return time, τ_{Ω_{ε}}^{K}(x), from Ω_{ε} into itself (see the precise definition in the next section), where Ω_{ε} = Ω_{ε}(z) is still a neighborhood of some point z ∈ X.

For the dynamical systems quoted above, a Poisson statistics can be proved, by showing that the distribution of successive return times into Ω_{ε} satisfies, for z μ-a.e.

μ_{Ω_{ε}} (x ∈ Ω_{ε} | τ_{Ω_{ε}}^{K}(x) ≤ t < τ_{Ω_{ε}}^{K+1}(x)) ⟶_{ε→0^{+}} \frac{t^{K}}{K!}e^{−t}  (2)

The preceding results deserve further investigations at least in two directions:

1. extend them to non hyperbolic dynamical systems and, more ambitiously, check their robustness when the system looses strong mixing properties.

2. prove an error estimate even for the distribution of successive return times (2) and relate this approximation rate, if possible, to the statistical properties of the system like correlations decay or spectral properties.

We try to give partial answer to these questions in this paper. The general setting we put in, is the return(s) times to the set Ω starting from itself, as expressed in formulas (1) and (2) (although in Theorem 2.1 we will also consider points starting everywhere). The first attempt was to give, for measure preserving dynamical systems, a general upper bound for the difference between the distribution of the (rescaled) first return time and the exponential-one law e^{−t} and then between the distribution of high-order (rescaled) return times and the Poisson law \frac{t^{K}}{K!}e^{−t}. We do not make any hypothesis on the set Ω, nor on the ergodic properties of μ; nevertheless these bounds are expressed in terms of the self-interactions of the set Ω and can be explicitly computed when typical rates of mixing are known (uniform mixing, α-mixing or φ-mixing). In this context, our bounds greatly improve and simplify the hypothesis of self-mixing condition of [Hir95], which was a powerful tool to get sufficient condition for the Poisson statistics. This first part of the paper is essentially due to one of us (B.S.) and is part of his Ph.D.

Thesis [Sau98b]. In the second part we apply the preceding bounds to new situations. The systems we treat are some non-uniformly hyperbolic maps of the interval; these maps are characterized by a structure parameter, say α, which measures the order of tangency at a neutral fixed point and governs the algebraic decay of correlations (in our example the order is n^{1−1/α}). If μ denotes the absolutely continuous invariant measure, we prove Poisson statistics (in the sense precise above), by giving an explicit approximation of the asymptotic law in terms of the measure of the set Ω_{n}, where in this case Ω_{n} is a decreasing sequence of cylinder sets chosen around almost all points in the interval. To be precise the error is of the type: μ(Ω_{n})^{β}, for any β < 1 − α, and therefore β is explicitly related to α and optimized just by 1 − α. For the distributions of the Kth return times the bounds simply become μ(Ω_{n})^{β/K}.

By inspecting these results, we could argue that the non-hyperbolic character of the maps reflects in the error term; to be more precise we think that as soon as the degree of non-uniform hyperbolicity of the map is monitored by a structure parameter α, this parameter will appear explicitly in the approximation to the Poisson law, which suggests, on the converse, that we could use Poissonian statistics to test lack of hyperbolicity. Our claim is motivated by two more observations: first, in getting these bounds we proved a sort of α-mixing for the map with a rate which was exactly the same as the algebraic rate for the correlations’ decay. Second, in the forthcoming paper [Sau98a] the return times is analyzed for a class of piecewise expanding multidimensional maps. Although the mixing properties are much more difficult to handle with, especially for the presence of singularity lines and the geometry of their shape, the uniform dilatation will provide bounds on the form: μ(Ω_{n})^{β} and μ(Ω_{n})^{β/K} for all β < 1, which reflects the fact that all the quantities involved, and the correlations’ decay too, admit exponential estimates. We will come back to these questions in section 4. As a final remark, we address two questions:

1. Our analysis is local: the events are chosen around almost all points which we could call, following a widespread tradition, generic (for our statistics). What happens if we consider non-generic points (discarding of course some trivial situation like fixed points) ? Could we see their (possibly different) statistics by involving some sort of large deviation argument ?

2. What is the place of Poissonian statistics regarding other ergodic characterizations of dynamical systems ? For example: what is the largest class of ergodic dynamical systems enjoying a Poissonian statistics ? Conversely, does an invariant measure satisfying that behavior implies strong ergodic properties too ?

2 General bounds on the distribution of return times

We will consider in this section a probability space (X, B, μ) together with a measure preserving transformation T acting on X. The basic object will be the return time into a positive measure set U starting from U defined by

τ_{U}(x) = inf {k ≥ 1| T^{k}x ∈ U} ∪ {∞}.

We define as usual the conditional measure μ_{U} on U by μ_{U}(A) = \frac{μ(A ∩ U)}{μ(U)}.

We then recall Kac’s theorem which says that the conditional expectation of τ_{U} given U is finite, and equal to 1/μ(U), when μ is ergodic. As indicated in the introduction, Kac’s result suggests how to properly rescale the return time when we are interested in its distribution.

2.1 First return time

We begin to show that the distribution of the first return time into the set U starting from U is close to an exponential one law if and only if the two distributions of the first return time starting, respectively from U and everywhere, are close.

Theorem 2.1. Let us define c(k, U) = μ_{U}(τ_{U} > k) − μ(τ_{U} > k) and set c(U) = sup_{k} |c(k, U)|. The distribution of the (rescaled) first return time into the set U differs from the exponential-one law by at most d(U) := 4μ(U) + c(U)(1 + log c(U)^{−1}), namely:

sup_{t≥0} |μ (τ_{U} > \frac{t}{μ(U)}) − e^{−t}| ≤ d(U)

which is still true starting from U:

sup_{t≥0} |μ_{U} (τ_{U} > \frac{t}{μ(U)}) − e^{−t}| ≤ d(U).

Conversely, the difference between the two distributions (starting inside U and everywhere) can be bounded in terms of the distance c̃(U) := sup_{t≥0} |μ_{U}(τ_{U} > t/μ(U)) − e^{−t}|, precisely:

c(U) ≤ 2μ(U) + c̃(U)(2 + log c̃(U)^{−1}).

Remark 2.2. Whenever μ(U) > 0 the return time’s law is discrete and this allow us to get a lower bound for the rate of convergence. More precisely, we have the following proposition:

Proposition 2.3. for each k ≥ 0

ε_{k,U} := |μ (τ_{U} > k) − e^{−kμ(U)}| + |μ (τ_{U} > k + 1/2) − e^{−(k+1/2)μ(U)}| ≥ \frac{e^{−kμ(U)}}{4} μ(U).

In particular, ε_{0,U} ≥ μ(U)/4.

Proof. [Proposition 2.3] Let k ≥ 0 be an integer. Since τ_{U} takes only integer values, the distribution for t = kμ(U) and t′ = (k + 1/2)μ(U) is the same, then

ε_{k,U} ≥ |exp(−kμ(U)) − exp(−(k + 1/2)μ(U))|
≥ exp(−kμ(U))(1 − e^{−μ(U)/2})
≥ \frac{e^{−kμ(U)}}{4} μ(U).

Proof. [Theorem 2.1] Let us remark that for any k ≥ 1 we have

μ(τ_{U} = k) = μ(U ∩ {τ_{U} > k − 1}).  (3)

Since {τ_{U} > k} = T^{−1}(U^{c} ∩ {τ_{U} > k − 1}) by the invariance of μ we get that μ(τ_{U} > k) = μ(τ_{U} > k − 1) − μ(U ∩ {τ_{U} > k − 1}), whence the result. Next, for all k > 0 we have

μ(τ_{U} > k) = μ(τ_{U} > k − 1) − μ(U)μ_{U}(τ_{U} > k − 1)
= μ(τ_{U} > k − 1) − μ(U)[μ(τ_{U} > k − 1) + c(k, U)]
= μ(τ_{U} > k − 1)[1 − μ(U)] − μ(U)c(k, U).

Then it follows by an immediate induction that

μ(τ_{U} > k) = (1 − μ(U))^{k} − μ(U) ∑_{j=1}^{k} c(j, U)(1 − μ(U))^{k−j}.

Hence for all t ≥ 0, putting k_{t} = [t/μ(U)], we have

|μ(τ_{U} > k_{t}) − (1 − μ(U))^{k_{t}} | ≤ μ(U) ∑_{j=1}^{k_{t}} |c(j, U)| ≤ tc(U).  (4)

Setting z = − log c(U), and k_{z} = [z/μ(U)], we get

(1 − μ(U))^{k_{z}} ≤ e^{−k_{z}μ(U)} ≤ c(U)e^{μ(U)} ≤ c(U) + 2μ(U),

for any t > z,

μ(τ_{U} > k_{t}) ≤ μ(τ_{U} > k_{z})
≤ (1 − μ(U))^{k_{z}} + zc(U)
≤ 2μ(U) + c(U)(1 − log c(U))

which gives |μ(τ_{U} > k_{t})−(1−μ(U))^{k_{t}} | ≤ 2μ(U)+c(U)(1−log c(U)). Instead for any t ≤ z the same estimate holds by inequality (4). Since, by an easy computation

|(1 − μ(U))^{k_{t}} − e^{−t}| ≤ 2μ(U),

we get for any t ≥ 0

|μ(τ_{U} > k_{t}) − e^{−t}| ≤ 4μ(U) + c(U)(1 − log c(U))

which proves the first part of the theorem. Moreover, since

|μ_{U}(τ_{U} > k_{t}) − μ(τ_{U} > k_{t})| = |c(k_{t}, U)| ≤ c(U),

we finally have for each t ≥ 0

|μ_{U}(τ_{U} > k_{t}) − e^{−t}| ≤ 4μ(U) + c(U)(2 − log c(U)).

The converse part is proven in the same way. For k ≥ 1,

μ(τ_{U} > k) = 1 − μ(τ_{U} ≤ k)
= 1 − ∑_{j=1}^{k} μ(τ_{U} = j)

= 1 − μ(U) ∑_{j=1}^{k} μ_{U}(τ_{U} > j − 1),

where we used in the last equality the relation (3). Hence

|μ(τ_{U} > k) − e^{−kμ(U)}| ≤ |1 − μ(U) ∑_{j=1}^{k} e^{−(j−1)μ(U)} − e^{−kμ(U)}| + kμ(U)c̃(U)

≤ |1 − μ(U)\frac{1 − e^{−kμ(U)}}{1 − e^{−μ(U)}} − e^{−kμ(U)}| + kμ(U)c̃(U)
≤ (1 + e^{−kμ(U)}) |1 − \frac{μ(U)}{1 − e^{−μ(U)}} | + kμ(U)c̃(U)
≤ 2μ(U) + kμ(U)c̃(U).

This gives, whenever k ≤ k_{0} := log c̃(U)^{−1}/μ(U):

|c(k, U)| ≤ 2μ(U) + c̃(U) log c̃(U)^{−1}.

For k > k_{0} we simply have

|c(k, U)| ≤ μ(τ_{U} > k_{0})+μ_{U}(τ_{U} > k_{0}) ≤ 2μ(U)+c̃(U) log c̃(U)^{−1}+e^{−k_{0}μ(U)}+c̃(U).

The last theorem gives a necessary and sufficient condition to obtain the exponential law, that is d(U) → 0. However, such a quantity is not very transparent for dynamical systems, that is why we give a criterion to estimate it. This kind of condition is a generalization of the so called “selfmixing condition” introduced in [Hir95].

Lemma 2.4. Let U ⊂ X a measurable set. The following estimate holds

c(U) ≤ inf {a_{N}(U) + b_{N}(U) + Nμ(U)| N ∈ ℕ}

where the quantities are defined by

a_{N}(U) = μ_{U}(⋃_{j=1}^{N} T^{−j}U) = μ_{U}(τ_{U} ≤ N),
b_{N}(U) = sup_{V ∈U_{∞}} |μ_{U}(T^{−N}V) − μ(V)|

with U = {U, U^{c}}, U_{n} = ⋁_{k=0}^{n−1} T^{−k}U and U_{∞} = ∪_{n}σ(U_{n}).

Proof. Let N ∈ ℕ. If k < N, we just bound c(k, U) by

|μ_{U}(τ_{U} > k) − μ(τ_{U} > k)| = |μ_{U}(τ_{U} ≤ k) − μ(τ_{U} ≤ k)|
≤ |μ_{U}(τ_{U} ≤ k)| + |μ(τ_{U} ≤ k)|
≤ a_{N}(U) + kμ(U) ≤ a_{N}(U) + Nμ(U).

Otherwise, let us remark that

{τ_{U} > k} and {τ_{U} ∘ T^{N} > k − N} differ only on {τ_{U} ≤ N}, and by hypothesis

|μ_{U}(τ_{U} > k) − μ_{U}(τ_{U} ∘ T^{N} > k − N)| ≤ μ_{U}(τ_{U} ≤ N) = a_{N}(U).

Moreover

|μ_{U}(τ_{U} ∘ T^{N} > k − N) − μ(τ_{U} > k − N)| =
|μ_{U}(T^{−N}(τ_{U} > k − N)) − μ(τ_{U} > k − N)| ≤ b_{N}(U).

But {τ_{U} > k − N} and {τ_{U} > k} differs only on {τ_{U} ∘ T^{k−N} ≤ N}, hence

|μ(τ_{U} > k − N) − μ(τ_{U} > k)| ≤ μ(τ_{U} ∘ T^{k−N} ≤ N) = μ(τ_{U} ≤ N) ≤ Nμ(U).

We finally get for each k, N ∈ ℕ

|μ_{U}(τ_{U} > k) − μ(τ_{U} > k)| ≤ a_{N}(U) + b_{N}(U) + Nμ(U)

which concludes the proof, since N is arbitrary.

We remark that b_{N}(U) is bounded by α(N) if the partition U = {U, U^{c}} is α-mixing, and by γ(N) if it is uniformly mixing (see definition 2.1 below). To simplify, we could say that the exponential law holds when there exists some N so small that only few points of U come back in U before N steps, but large enough such that T^{N}U is uniformly spread out.

Definition 2.1 (Speed of mixing). Let (X, B, T, μ) be a dynamical system and ξ a finite or countable measurable partition of X. We set ξ_{k} = ⋁_{j=0}^{k−1} T^{−j}ξ and σ(ξ_{k}) the σ-algebra generated by ξ_{k}.

1. Uniform mixing. The partition ξ is uniformly mixing with speed γ(n) going to zero for n going to infinity if for any n,

γ(n) = sup_{k,l} sup_{\begin{matrix}R∈σ(ξ_{k}) \\ S∈T^{−(n+k)}σ(ξ_{l}) \end{matrix}} |μ(R ∩ S) − μ(R)μ(S)|.

2. α-mixing. The partition ξ is α-mixing with speed α(n) going to zero for n going to infinity if for any n,

α(n) = sup_{k,l} sup_{\begin{matrix}R∈ξ_{k} \\ S∈T^{−(n+k)}σ(ξ_{l}) \end{matrix}} |\frac{μ(R ∩ S)}{μ(R)} − μ(S)| .

3. φ-mixing. The partition ξ is φ-mixing with speed φ(n) going to zero for n going to infinity if for any n,

φ(n) = sup_{k,l} sup_{\begin{matrix}R∈σ(ξ_{k}) \\ S∈T^{−(n+k)}ξ_{l} \end{matrix}} |\frac{μ(R ∩ S)}{μ(R)μ(S)} − 1| .

4. Weak-Bernoulli. The partition ξ is weak-Bernoulli with speed β(n) going to zero when n goes to infinity, if for any n,

β(n) = sup_{k,l} ∑_{\begin{matrix}R∈ξ_{k} \\ S∈T^{−(n+k)}ξ_{l} \end{matrix}} |μ(R ∩ S) − μ(R)μ(S)|.

Remark 2.5. We state some general implications and results verified by the preceding types of mixing.

1. φ-mixing implies α-mixing which implies uniform mixing. For any n, γ(n) ≤ α(n) ≤ φ(n).

2. φ-mixing implies weak-Bernoulli which implies uniform mixing. For any n, γ(n) ≤ β(n) ≤ φ(n).

3. If ξ is a generating partition of an uiformly mixing dynamical system, then the system is mixing.

4. If ξ is a generating weak-Bernoulli partition then the system is metrically conjugated with a Bernoulli shift.

2.2 Successive return times

We will now investigate the properties of successive return times to the set U. For this purpose, let us define the kth return time in U by

τ_{U}^{(k)}(x) = \begin{cases}0 & if k = 0, \\ τ_{U}(x) + τ_{U}^{(k−1)}(T^{τ_{U} (x)}(x)) & if k > 1. \end{cases}

Observe that the difference between two consecutive return times follows the same law than the first, for the simple reason that

τ_{U}^{(K+1)} − τ_{U}^{(K)} = τ_{U} ∘ T^{τ_{U}^{(K)}}

and the measure μ_{U} is invariant with respect to the induced application on U.

Theorem 2.6. Let U ⊂ X be a measurable set, and U = {U, U^{c}} the partition associated to it. Given an integer K and a rectangle Q_{K} in ℝ^{K}, the differences between successives normalized return times in U are independent and exponentially distributed up to f(K, U) (see (5) below), where f(K, U) is defined depending on the type of mixing by

(α) When (X, T, μ) is α-mixing for U, with speed α ^{1}, then

f(K, U) = K (3d(U) + inf_{M∈ℕ}{α(M) + 3Mμ(U)}) .

α′(N) = sup \{|\frac{μ(R ∩ S)}{μ(R)} − μ(S)| | j, N ∈ ℕ, R ∈ U_{j}, T^{j}R ⊂ U, V ∈ T^{−j−N}U_{∞}\}.

1We just need that mixing property for some special sets, more precisely, we are interested by

(γ) When the partition U is uniformly mixed by (X, T, μ) with speed γ, then

f(K, U) = K [4d(U) + inf_{\begin{matrix}M∈ℕ \\ γ(M)<μ(U)^{2} \end{matrix}} \{\frac{γ(M)}{μ(U)^{2}} (2 − K log \frac{γ(M)}{μ(U)^{2}} ) + 3Mμ(U)\}] .

Indeed the following inequality holds:

|μ_{U} ((τ_{U}^{(1)}, τ_{U}^{(2)}−τ_{U}^{(1)}, …, τ_{U}^{(K)}−τ_{U}^{(K−1)}) ∈ \frac{1}{μ(U)}Q_{K}) − ∫_{Q_{K}} ∏_{i=1}^{K} e^{−s_{i}}ds^{K} | ≤ f(K, U). (5)

Remark 2.7. Note that the mixing assumption is made only for the special partition U. If the system has a partition Z (not necessarily with two elements), uniformly mixing with speed γ_{Z}, then for any cylinder U ∈ Z_{n} of order n, the partition U = {U, U^{c}} is still uniformly mixing with speed γ_{U}(M) ≤ γ_{Z}(M − n). The proof of the theorem is inspired by [CG93], with the following differences: 1) U is any measurable set; 2) we take care of the approximations to get an estimation of the error; 3) we still get an estimation even if the system is uniformly mixing; however, it is interesting whenever γ(M) = o(1/M^{2}).

Proof. [Theorem 2.6] Let us remark first that if we denote by F = T^{τ_{U}} the induced application on U, then for each k ∈ ℕ,

τ_{U}^{(k+1)} − τ_{U}^{(k)} = τ_{U} ∘ F^{k}.

We set τ_{k} = (τ_{U}, τ_{U} ∘ F, ⋯, τ_{U} ∘ F^{k−1}). We will show that the inequality (5) holds by induction on K.

For K = 1, we apply Theorem 2.1 which gives, setting Q_{1} = [u, v],

|μ_{U}(τ_{U} ∈ [u, v]) − ∫_{u}^{v} e^{−s}ds| = |μ_{U}(τ_{U} > v) − μ_{U}(τ_{U} > u) − (e^{−u} − e^{−v})| ≤ 2d(U).

Let’s suppose that the inequality (5) is true for K; we want to prove that it is also true for K + 1. Let [r, s] be the projection of Q_{K+1} onto the last coordinate, and for k = K, K + 1 denote:

D_{k} = U ∩ τ_{k}^{−1}(\frac{1}{μ(U)}Q_{k}).

For any M ∈ ℕ, the set defined by

E_{K+1}(M) = D_{K} ∩ {x ∈ U| τ_{U} ∘ T^{M} ∘ F^{K}(x) ∈ [r, s]/μ(U) − M} verifies the inclusions

E_{K+1}(M) ∩ {τ_{U} ∘ F^{K} > M} ⊂ D_{K+1} ⊂ E_{K+1}(M) ∪ {τ_{U} ∘ F^{K} ≤ M}. Theorem 2.1 shows that the two sets which bound D_{K+1} do not differ too much, namely, μ_{U}(τ_{U} ∘ F^{K} ≤ M) = μ_{U}(τ_{U} ≤ M) ≤ 1 − e^{−Mμ(U)} + d(U) ≤ Mμ(U) + d(U). Therefore we get the first bound

|μ_{U}(D_{K+1}) − μ_{U}(E_{K+1}(M))| ≤ Mμ(U) + d(U).  (6)

So the problem reduces to prove that μ_{U}(E_{K+1}(M)) follows the expected law. We decompose the sets E_{K+1}(M) over A_{K}^{j} = U ∩ {τ_{U}^{(K)} = j}. We have

E_{K+1}(M) ∩ A_{K}^{j} = D_{K} ∩ A_{K}^{j} ∩ T^{−(M+j)}\{τ_{U} ∈ \frac{[r, s]}{μ(U)} − M\}.

We can now use the mixing with R = D_{K}∩A_{K}^{j} ∈ σ(U_{j}) and S = T^{−(M+j)}{τ_{U} ∈ [r, s]/μ(U) − M}. According to the type of mixing, we get two approximations:

(α) When the partition U is α-mixing:

|μ_{U}(E_{K+1}(M) ∩ A_{K}^{j}) − μ_{U}(D_{K} ∩ A_{K}^{j})μ(τ_{U} ∈ \frac{[r, s]}{μ(U)} − M)| ≤ α(M)μ_{U}(D_{K} ∩ A_{K}^{j}).

Summing over the possible values of j we get:

|μ_{U}(E_{K+1}(M)) − μ_{U}(D_{K})μ(τ_{U} ∈ \frac{[r, s]}{μ(U)} − M)| ≤ α(M)μ_{U}(D_{K}) ≤ α(M).(7)

Now Theorem 2.1 gives

|μ(τ_{U} ∈ \frac{[r, s]}{μ(U)} − M) − (e^{−r} − e^{−s})| ≤ |μ(τ_{U} ∈ \frac{[r, s]}{μ(U)}) − (e^{−r} − e^{−s})| + 2Mμ(U)
≤ 2(Mμ(U) + d(U)).

We briefly recall the approximations done with their respective errors

μ_{U}(D_{K+1}) → μ_{U}(E_{K+1}(M)) → μ_{U}(D_{K})μ\{τ_{U} ∈ \frac{[r,s]}{μ(U)}\} → μ_{U}(D_{K})(e^{−r} − e^{−s})
↓  ↓  ↓
Mμ(U) + d(U)  α(M)  2(Mμ(U) + d(U))

This allows us to show that the difference

|μ_{U}(D_{K+1}) − ∫_{Q_{K+1}} ∏_{i=1}^{K+1} e^{−s_{i}}ds^{K+1}|  (8)

is bounded by the quantity f(K, U)+3Mμ(U)+α(M)+3d(U) ≤ f(K+1, U), which proves the induction and concludes the proof of this first case.

(γ) We now consider the case when U is uniformly mixing: Let M be such that γ(M) < μ(U)^{2}. As a first step, we can restrict ourselves

to the case when Q_{K} ⊂ [0, z]^{K}, with z = − log \frac{γ(M)}{μ(U)^{2}} > 0. In fact,

Q_{K} \ [0, z]^{K} ⊂ ⋃_{k=1}^{K} ℝ_{+}^{k−1}×]z, ∞] × ℝ_{+}^{K−k}

which implies using Theorem 2.1

μ_{U}(μ(U)τ_{K} ∈ Q_{K} \ [0, z]^{K}) ≤ ∑_{k=1}^{K} μ_{U}(τ_{U}^{(k+1)} − τ_{U}^{(k)} > z/μ(U))
= Kμ_{U}(τ_{U} > z/μ(U))
≤ K(e^{−z} + d(U)).

Moreover

∫_{Q_{K}\[0,z]^{K}} ∏_{i=1}^{K} e^{−s_{i}}ds^{K} ≤ ∑_{k=1}^{K} ∫_{ℝ_{+}^{k−1}×]z,∞]×ℝ_{+}^{K−k}} ∏_{i=1}^{K} e^{−s_{i}}ds^{K} ≤ Ke^{−z}.

Next, by decomposing according to

μ_{U}(μ(U)τ_{K} ∈ Q_{K}) = μ_{U}(μ(U)τ_{K} ∈ Q_{K} ∩ [0, z]^{K}) + μ(μ(U)τ_{K} ∈ Q_{K} \ [0, z]^{K}), we get f(K, U) ≤ K(2e^{−z} +d(U))+f′(K, U) where f′(K, U) is the maximum of the difference (5) for the boxes Q_{K} ⊂ [0, z]^{K}. We then estimate f′(K, U). First by uniform mixing we get

|μ_{U}(E_{K+1}(M) ∩ A_{K}^{j}) − μ_{U}(D_{K} ∩ A_{K}^{j})μ(τ_{U} ∈ [r, s]/μ(U) − M)| ≤ \frac{γ(M)}{μ(U)}

and then we sum over all possible^{2} values j of τ^{(K)},

|μ_{U}(E_{K+1}(M)) − μ_{U}(D_{K})μ(τ_{U} ∈ [r, s]/μ(U) − M)| ≤ \frac{Kzγ(M)}{μ(U)^{2}}.

The same computation performed after estimation (7) (where now α(M) is replaced by Kzγ(M)/μ(U)^{2} in inequality (7)), gives the bound f′(K+1, U) ≤

K \frac{zγ(M)}{μ(U)^{2}} + 3(d(U) + Mμ(U)). Then for each M,

f′(K, U) ≤ K^{2} \frac{zγ(M)}{μ(U)^{2}} + 3K(d(U) + Mμ(U)).

Since M is arbitrary, our choice of z implies that the inequality (5) is verified with

f(K, U) = K [4d(U) + inf_{\begin{matrix}M∈ℕ \\ γ(M)<μ(U)^{2} \end{matrix}} \{\frac{γ(M)}{μ(U)^{2}} (2 − K log \frac{γ(M)}{μ(U)^{2}} ) + 3Mμ(U)\}] .

We are now ready to give the most important result of this section, namely, to prove the Poisson statistics for successive return times. Let N(t) be the number of visits into U up to the normalized time t/μ(U),

N(t) = sup {K > 0| τ_{U}^{(K)} ≤ t/μ(U)}.

It turns out that N(t) is a discrete random variable whose law is close to a Poissonian one, more precisely we have Theorem 2.8. The distribution of the number of visits N(t) differs from the Poissonian law by

2Since Q_{K} ⊂ [0, z]^{K}, the Kth return time is less or equal to Kz, hence it takes at most [Kz] different values.

|μ_{U}(N(t) = K) − \frac{t^{K}}{K!}e^{−t}| ≤ g(t, K, U) + g(t, K + 1, U),

where for each k ≥ 0 g(t, k, U) = (12t^{k}/k + k^{k−1}) \sqrt[k]{f(k, U)}.

Proof. It is a consequence of the weak dependence of the differences of successives return times established by Theorem 2.6. We first remark that

μ_{U}(N(t) = K) = μ_{U} (\{τ_{U}^{(K)} ≤ \frac{t}{μ(U)}\} ∩ \{τ_{U}^{(K+1)} > \frac{t}{μ(U)}\})
= μ_{U} (τ^{(K)} ≤ t/μ(U)) − μ_{U} (τ^{(K+1)} ≤ t/μ(U)) .

It is then sufficient to compute the measure of points whose kth rescaled return time is smaller than t, for k = K, K+1. If we put P̃_{k}(t) the distribution of the sum of the differences of successives return times, we know that when the latter are i.i.d. random variables with the same exponential law, then setting L_{k}(t) = {(s_{1}, …, s_{k}) ∈ ℝ_{+}^{k}| s_{1} + ⋯ + s_{k} ≤ t} we get

P̃_{k}(t) = P_{k}(t) := ∫_{L_{k}(t)} ∏_{i=1}^{k} e^{−s_{i}}ds_{i}

which gives the classical result P_{K}(t) − P_{K+1}(t) = \frac{t^{K}}{K!}e^{−t}.

The difficulty comes now from the fact that we have to translate Theorem 2.6 given for boxes on the simplex L_{k}(t).

Let’s suppose that f(k, U) < 1, otherwise there is nothing to prove. Hence the integer defined by N = [k/f(k, U)^{k+1}] is bigger than k. We consider the uniform partition of [0, t]^{k} by cubes of size t/N. Let Δ_{k} be the union of those cubes Q_{k} included in the interior of L_{k}(t), for which for any (s_{1}, …, s_{k}) ∈ Q_{k}, ∑_{i=1}^{k} s_{i} < t and Σ_{k} those which intersect the boundary, i.e. the union of those cubes such that there exists (s_{1}, …, s_{k}) ∈ Q_{k} with ∑_{i=1}^{k} s_{i} = t. By using Figure 1: Partition of the cube [0, t]^{k} for k = 2. Σ_{k} is the union of dotted squares and Δ_{k} the union of shaded rectangles R_{k}(Q_{k}).

the notation τ_{k} introduced in the proof of Theorem 2.6 we have,

δ := |μ_{U}(τ_{U}^{(k)} ≤ t/μ(U)) − ∫_{L_{k}(t)} ∏_{i=1}^{k} e^{−s_{i}}ds^{k}|

≤ |μ_{U}(τ_{k} ∈ \frac{Δ_{k}}{μ(U)}) − ∫_{Δ_{k}} ∏_{i=1}^{k} e^{−s_{i}}ds^{k}| + μ_{U}(τ_{k} ∈ \frac{Σ_{k}}{μ(U)}) + ∫_{Σ_{k}} ∏_{i=1}^{k} e^{−s_{i}}ds^{k}
≤ δ_{1} + δ_{2} + δ_{3}.

To estimate δ_{1}, we put Π for the projection over the k−1 last coordinates; then the sets R_{k}(Q_{k}) = {Q′_{k} ∈ Δ_{k}|Π(Q′_{k}) = Π(Q_{k})} are boxes, and their number is bounded by N^{k−1} (see Figure 1). For each of these boxes Theorem 2.6 gives an error smaller than f(k, U), and then we get δ_{1} ≤ N^{k−1}f(k, U).

To compute δ_{2} and δ_{3}, we first remark that a straightforward combinatorial calculus gives, for the number C_{N}^{k} of cubes inside Σ_{k}, C_{N}^{k} ≤ 6N^{k−1} (see [Sau98b]). But for each cube Q_{k} ⊂ Σ_{k} Theorem 2.6 gives

μ_{U}(τ_{k} ∈ Q_{k}) ≤ ∫_{Q_{k}} ∏_{i=1}^{k} e^{−s_{i}}ds^{k} + f(k, U).

Summing over all the cubes contained in Σ_{k} one has δ_{2} ≤ 6N^{k−1}f(k, U) + δ_{3}.

Moreover the integral ∫_{Q_{k}} ∏_{i=1}^{k} e^{−s_{i}}ds^{k} is bounded by the volume of Q_{k} equal to (t/N)^{k}, which gives δ_{3} ≤ 6N^{k−1}t^{k}/N^{k}. We then deduce that

δ ≤ δ_{1} + δ_{2} + δ_{3} ≤ N^{k−1}f(k, U) + 12t^{k}/N

which implies δ ≤ (12t^{k}/k + k^{k}) f(k, U) by the previous choice of N.

3 Applications

In the preceding chapter we gave general estimates for the error between the distribution of the number of visits into a set U and the Poissonian law. We could wonder whether this law is attained in the limit of μ(U) → 0. Put in this way the question is not very clear. What we need is instead to localize a sequence of neighborhoods U_{ε}(z) shrinking to zero and ask whether the Poisson law holds in the limit ε → 0. This approach was successfully carried out by several authors as reminded in the introduction. Although their results were applied to dynamical systems, the inspiration and some of the techniques of the proofs were of probabilistic nature (theory of moments, Laplace transform). Here we follow a purely dynamical direction, trying to extract all the statistical informations by the ergodic properties of the system. In this way we are able, for example, to exhibit the Poissonian statistics for a large class of non uniformly hyperbolic maps of the interval, widely studied in the last years especially to determine the rate of decay of correlations and the central limit theorem.

Some statistical properties of these maps have been studied in the paper [LSV97] (this paper contains a quite complete bibliography on the subject), where an absolutely continuous invariant probability measure (acim) is first constructed, and then it is shown that it enjoys a polynomial decay of correlations.

One feature of these maps is that they are characterized by a structure parameter (the order of tangency at an indifferent fixed point), which governs the statistical properties, and that can be viewed as an indicator of the “weak” hyperbolicity of the map. Actually, it turns out that this parameter appears even in the approximation to the Poissonian law.

Let’s then consider for 0 < α < 1 the following map of the unit interval:

T (x) = \begin{cases}x(1 + 2^{α}x^{α}) & ∀x ∈ [0, 1/2) \\ 2x − 1 & ∀x ∈ [1/2, 1] \end{cases} .

We recall some properties and results which we will need in the following, and we refer the reader to the quoted paper for more informations and proofs. This application has a finite Markov partition (with two elements), but for our purposes it is more convenient to work with the countable one ξ generated by the left preimages a_{n} of 1, ξ = {A_{m}| m ∈ ℕ} with A_{n} =]a_{n+1}, a_{n}]. We will often use in the following the easy bound \frac{a_{n}}{a_{n+1}} ≤ 2.

We can associate to each point z ∈ X =]0, 1] an unique infinite sequence \underline{ω} = ω_{1}ω_{2}... with the property that T^{m−1}z ∈ A_{ω_{m}} for all integer m ≥ 1. We denote by ξ_{m} the dynamical partition ξ ∨ T^{−1}ξ ⋯ T^{−m+1}ξ and call its elements m-cylinders. We denote with ξ_{m}(z) ∈ ξ_{m} the m-cylinder which contains z. The sequence \underline{ω} satisfies the admissibility condition: ω_{m}ω_{m+1} appears in \underline{ω} if and only if ω_{m} = 0 or ω_{m+1} = ω_{m} − 1. We say that a non empty cylinder C = [ω_{1} … ω_{k}] ∈ ξ_{k} is maximal if it is maps onto X after exactly k iterations, which is easily seen to be equivalent to ω_{k} = 0.

3.1 Some mixing properties

We begin with a brief surveys of some results proved by two of us (B.S., S.V) in the joint paper [LSV97] with Carlangelo Liverani. We showed that the density h of the acim belongs to a certain cone of functions C_{∗}(a), which will be characterized later (see Lemma 3.2), provided a is big enough, and satisfies ^{3}:

Lemma A (Lemma 2.2 in [LSV97]) The cone C_{∗}(a) is left invariant by the Perron-Frobenius operator P, i.e. P (C_{∗}(a)) ⊂ C_{∗}(a).

Lemma B (Lemma 2.3 in [LSV97]) The density h belongs to the cone C_{∗}(a), and verifies in particular whenever x ≤ y

\frac{h(x)}{h(y)} ≤ (y/x)^{α+1},  (9)

h(x) ≤ ax^{−α}.  (10)

P f(x) = ∑_{T y=x} \frac{1}{D_{y}T} f(y).

3We recall the formal definition of the Perron Frobenius operator P acting on function f : [0, 1] → ℝ:

One easily check that μ is an acim iff h = \frac{dμ}{dx} is a fixed point of P on L^{1}(dx).

Proposition C (Distortion inequality, proof of Proposition 3.3 in [LSV97]) There exists some constant Δ such that for all k and x, y ∈ C ∈ ξ_{k},

\frac{D_{x}T^{k}}{D_{y}T^{k}} ≤ Δ < ∞.  (11)

We will suppose without loss of generality that a ≥ 4Δ.

Theorem D (Theorem 4.1 in [LSV97]) In the proof of this theorem we in particular got that for f ∈ C_{∗}(a),

∥P^{n}(f − λ(f))∥_{L^{1}(λ)} ≤ Φ(n)∥f∥_{L^{1}(λ)}  (12)

with Φ(n) = Cn^{− \frac{1}{α} +1}(log n)^{\frac{1}{α}} = O_{L}(n^{− \frac{1}{α} +1}), where we define by O_{L}(ε) = O(ε(log ε^{−1})^{r}) in the limit ε → 0, for any constant r.

We then need a few more results on the speed of mixing which turn out to be useful for the statistics of return times and also to establish the weak-bernoullicity of the map.

Lemma 3.1. For any z ∈ X, and for any m such that ξ_{m}(z) is maximal, the partition U = {ξ_{m}(z), ξ_{m}(z)^{c}} satisfies a property close to the α-mixing, namely

α′(N) = sup_{j∈ℕ} sup_{\begin{matrix}R∈U_{j} \\ T^{j}R⊂U \end{matrix}} sup_{S∈U_{∞}} |\frac{μ(R ∩ T^{−N−j}S)}{μ(R)} − μ(S)| = O_{L}((N − m)^{1− \frac{1}{α}}).

Proof. Let z be a point of X and m be an integer such that ξ_{m}(z) is maximal. Let U be the partition given by ξ_{m}(z) and its complement, and U_{j} the refinement of U. For R ∈ U_{j} such that T^{j}R ⊂ U, we have R ∈ σ(ξ_{m+j}) and R is a union of maximal cylinders V_{m+j}^{k} ∈ ξ_{m+j}; choose V ∈ ξ_{m+j} one of these maximal cylinders. For any S ∈ T^{−(N+j)}B there exists a set W ∈ B such that R = T^{−(N+j)}W. We then have

(∗) := μ(V ∩ S) − μ(V)μ(S)
= ∫ 1I_{V} 1I_{W} ∘ T^{N+j}hdλ − ∫ μ(V)h1I_{W} dλ
= ∫ P^{N+j}[h(1I_{V} − μ(V))]1I_{W} dλ
≤ ∥P^{N+j}[h(1I_{V} − μ(V))]∥_{L^{1}(λ)}.

By exploiting the fact that V is maximal we continue the preceding bound as

(∗) ≤ ∥P^{N−m}[P^{j+m}(h1I_{V}) − μ(V)]∥_{L^{1}(λ)} + ∥P^{N−m}[μ(V)h − μ(V)]∥_{L^{1}(λ)}
≤ 4aΦ(N − m)μ(V),

with Φ given by inequality (12), provided P^{m+j}(h1I_{V}) ∈ C_{∗}(a), which is the case by the Lemma 3.2 below. We conclude the proof by summing over all the maximal cylinders of R.

Lemma 3.2. For any maximal cylinder V ∈ ξ_{p},

P^{p}(h1I_{V}) ∈ C_{∗}(a).

Proof. We first set f := P^{p}(h1I_{V}) and T_{V}^{p} : V → X the restriction of T^{p} to V. Since T^{p} is injective over V we can rewrite f as

f(x) = h ∘ T_{V}^{−p}(x)D_{x}T_{V}^{−p}

which in particular shows that f is continuous. To prove that f belongs to the cone of smooth functions C_{∗}(a) we must verify the following four properties which just define the cone.

1. f is continuous and positive, that is clear in our case.

2. f is decreasing. Since h ∈ C_{∗}(a), h decreases. In addition, T_{V}^{−p} is decreasing and concave, therefore h ∘ T_{V}^{−p} and DT_{V}^{−p} decrease.

3. x ↦ x^{α+1}f(x) increases. Since T_{V}^{−p} : X → V is increasing, an equivalent statement is that

(T^{p}u)^{α+1}h(u)\frac{1}{D_{u}T^{p}}

is increasing with u ∈ V. Observing that

(\frac{T^{p}u}{u})^{α+1} \frac{1}{D_{u}T^{p}}

increases over V ∈ ξ_{p} (which is true for p = 1 and the general case is proved by recurrence), and u ↦ u^{α+1}h(u) increases, we obtain the result.

4. f(x) ≤ ax^{−α} ∫ f. Since f is continuous, there exists v ∈ V such that

∫ f = f(T^{p}v) = h(v)\frac{1}{D_{v}T^{p}} .

The distortion estimate (11) for u ∈ V ∈ ξ_{p} gives

\frac{D_{v}T^{p}}{D_{u}T^{p}} ≤ Δ.

Moreover since h decreases, inequality (9) yields

\frac{h(u)}{h(v)} ≤ \frac{h(a_{ω_{1}+1})}{h(a_{ω_{1}})} ≤ (\frac{a_{ω_{1}}}{a_{ω_{1}+1}} )^{α+1} ≤ 4.

As a consequence, we get for u = T_{V}^{−p}x

f(x) = h(u)\frac{1}{D_{u}T^{p}} ≤ 4h(v)\frac{Δ}{D_{v}T^{p}} ≤ ax^{−α} ∫ f,

because x ≤ 1 and 4Δ ≤ a.

We finally prove that the countable partition ξ, and therefore the twoelements one, is weakly Bernoulli.

Theorem 3.3. The partition ξ is weakly Bernoulli for (X, T, μ) with speed β(n) = O_{L}(n^{1−1/α}).

Proof. We begin to recall the following result by Hofbauer and Keller [HK82] which permits to bound β(n) as

β(n) ≤ sup_{m∈ℕ} ∑_{R∈ξ_{m}} ∥P^{n+m}((1I_{R} − μ(R))h)∥_{L_{λ}^{1}}.  (13)

Then it will be enough to bound

∥P^{m+n}((1I_{R} − μ(R))h)∥

with R ∈ ξ_{m}. Let p_{R} ≥ m be the integer for which R ∈ ξ_{p_{R}} is maximal. We decompose the sum over all the cylinders R ∈ ξ_{m} into two blocks. Let M(m, n) be the set of maximal cylinders for p_{R} < m + n/2. When R ∈ M(m, n), the same computation performed in Lemma 3.1 gives

∥P^{m+n}((1I_{R} − μ(R))h)∥_{L_{λ}^{1}} ≤ μ(R)O_{L}((m + n − p_{R})^{1−1/α}) = μ(R)O_{L}(n^{1−1/α}).

Then the set of cylinders which do not belong to M(m, n) is exactly T^{−m+1}[0, a_{n/2}], whose measure is equal to

μ(T^{−m+1}[0, a_{n/2}]) = μ([0, a_{n/2}]) = ∫_{0}^{a_{n/2}} h(x)dx = O(n^{1−1/α}).

This proves the theorem.

3.2 Statistics of return times

We now come back to the study of return times and the first step will be the estimation of the quantities involved in the error term given by Lemma 2.4.

Lemma 3.4. There exists a constant B such that for any k and C ∈ ξ_{k} with T^{−k}C ∩ C ≠ ∅,

sup P^{k}1I_{C} ≤ Bk^{−1−1/α}.  (14)

Proof. Let k_{0} be such that D_{a_{k_{0}}}T ≤ 2, and put r = D_{a_{k_{0}}}T > 1. Let C = [ω_{1}...ω_{k}] be a k-cylinder such that T^{−k}C ∩ C ≠ ∅. This implies that ω_{k}ω_{1} is admissible. We want to estimate sup P^{k}1I_{C} = 1/ inf_{C} DT^{k}. If ω_{j} ≤ k_{0} for all j = 1..k, then DT^{k} ≥ r^{k}. Else, take j such that ω_{j} = max_{1≤i≤k} ω_{i}. Either j = 1, and consequently ω_{k} = 0 or ω_{j−1} = 0. In the last case we have

inf_{C} DT^{k} ≥ inf_{[ω_{1}...ω_{j−1}]} DT^{j−1} inf_{[ω_{j}...ω_{k}]} DT^{k+1−j} ≥ Δ^{−1} inf_{[ω_{j}...ω_{k}ω_{1}...ω_{j−1}]} DT^{k}.

By this argument we are led to consider the worst case which is given by a cylinder of type C = [(k − 1)(k − 2)...0]. For T^{k}C = [0, 1], the distortion formula (11) and the estimation a_{k} ≤ ck^{−1/α} given by Lemma 3.2 in [LSV97] we get D_{a_{k}}T^{k} = c′k^{1+1/α} for some constant c′, from which the lemma follows by taking B ≥ Δ/c′ such that Bk^{1−1/α} ≥ r^{k} for all k > 0.

We now introduce the first return time of a cylinder U which plays a crucial role in [Hir95]. We define it as τ(U) = inf {τ_{U}(x)| x ∈ U}.

Lemma 3.5. The quantity a_{N}(U) defined in Lemma 2.4 for U = ξ_{m}(z) is bounded by,

a_{N}(U) = \frac{4Δ}{inf h} \frac{Nμ(U)}{λ(T^{τ(U)}U)}.

Proof. We suppose N > τ(U) otherwise a_{N}(U) = 0. Set τ = τ(U); for each z in X we have

a_{N}(U) ≤ ∑_{j=1}^{N} \frac{1}{μ(U)}μ(T^{−j}U ∩ U)

= ∑_{j=τ}^{N} \frac{1}{μ(U)} ∫ P^{j}(1I_{U}h)1I_{U}dλ

≤ N sup_{j=τ..N} sup_{U} \frac{P^{j}(1I_{U}h)}{h}.

Now the distortion (11) and the regularity of the density (9) give

P^{τ}(1I_{U}h) = h ∘ T_{U}^{−τ}DT_{U}^{−τ}1I_{T^{τ} U}
≤ 4Δ\frac{1}{λ(T^{τ}U)} ∫_{T^{τ} U} h ∘ T_{U}^{−τ}DT_{U}^{−τ}1I_{T^{τ} U}dλ
≤ 4Δ\frac{μ(U)}{λ(T^{τ}U)}.

Finally, P h = h and since P is a positive operator one has

\frac{P^{j}(1I_{U}h)}{h} ≤ \frac{P^{j−τ}1I}{h} sup P^{τ}(1I_{U}h) ≤ \frac{P^{j−τ} \frac{h}{inf h}}{h} sup P^{τ}(1I_{U}h) ≤ \frac{4Δ}{inf h} \frac{μ(U)}{λ(T^{τ}U)}.

The next step will be to show that τ(U) is almost everywhere big enough to give a good upper bound in the previous lemma for a_{N}(U).

We first define in full generality the local rate of return for cylinders. As a matter of fact, we would like to point out that the first return time of a set into itself allows to define and compute an interesting dimension-like characteristic which we called the Afraimovich-Pesin dimension in [PSV98].

Definition 3.1. Let ζ a partition of X. Denote with ζ_{n}(x) the element of ζ ∨ T^{−1}ζ ∨ ⋯ ∨ T^{−n+1}ζ which contains x ∈ X. We then define the local (lower and upper) rate of return for cylinders as

\overline{\underline{R}}_{ζ}(x) = \overline{\underline{lim}}_{n→∞} \frac{τ(ζ_{n}(x))}{n}.

Proposition 3.6. (i) Both \underline{R}_{ζ} and \overline{R}_{ζ} are sub-invariant, namely \underline{R}_{ζ}∘T ≤ \underline{R}_{ζ} and \overline{R}_{ζ} ∘ T ≤ \overline{R}_{ζ}.

(ii) Assume that ζ is a measurable partition of the measurable space X, and μ is an invariant probability, then \underline{R}_{ζ} and \overline{R}_{ζ} are μ-a.e. invariant.

(iii) Moreover, whenever μ is ergodic \underline{R}_{ζ} and \overline{R}_{ζ} are μ-a.e. constant

Proof. (i) Let x ∈ X. For each integer n > 0, we have:

ζ_{n}(x) ∩ T^{k}ζ_{n}(x) ≠ ∅ ⟹ ζ_{n−1}(T x) ∩ T^{k}ζ_{n−1}(T x) ≠ ∅.

which implies that τ(ζ_{n−1}(T x)) ≤ τ(ζ_{n}(x)).

(ii) is a standard property of sub-invariant functions on finite measure spaces and then (iii) follows immediately.

We state the following result which can be improved for some subshifts^{4}.

Proposition 3.7. For μ-almost every z ∈ X, the lower rate of return for cylinders is equal to 1.

\underline{R}_{ξ}(z) = 1.

Proof. Let 1/2 < δ < 1. Consider the set (we denote N_{m}(z) = τ(ξ_{m}(z))),

L_{m} := {z ∈ A_{0}| N_{m}(z) ≤ δm}.

If

∑_{m=1}^{∞} μ(L_{m}) < ∞,  (15)

4We have in fact the following:

Theorem. Suppose that μ is a Gibbs state for the Hölder potential φ on some irreducible and aperiodic subshift of finite type with finite alphabet ζ, then μ-almost everywhere, \overline{R}_{ζ} = \underline{R}_{ζ} = 1.

Proof. An easiest version of the Proposition 3.7 gives the lower bound, while the uniform upper bound τ(C_{n}) ≤ n + n_{0} holds, where C_{n} is a cylinder of order n, and n_{0} is the lowest power for which the transition matrix becomes strictly positive.

then the Borel-Cantelli Lemma ensures that for almost every z ∈ A_{0}, we have N_{m} > δm, up to finitely many m. By sending δ to 1 we show that \underline{R}_{ξ}(z) ≥ 1 almost everywhere on A_{0}. Then for the preceding proposition (iii) and the ergodicity of the measure μ, we get the same bound almost everywhere. The equality finally follows since each time that T^{m−1}z ∈ A_{0}, we have T^{m}ξ_{m}(z) = X hence N_{m}(z) ≤ m.

In order to prove (15) it is sufficient to consider Lebesgue measure instead of μ (since the density h is bounded from below). We have

λ(L_{m}) = ∑_{k=1}^{[m/2]} λ(N_{m} = k) + ∑_{k=[m/2]+1}^{δm} λ(N_{m} = k).
(1) + (2)

We now perform a detailed analysis of the sets appearing in the preceding formula.

(1): In this case, the cylinder ξ_{m}(z) with N_{m} = k must be of the form

ξ_{m}(z) = [(ω_{1}.. \underbrace{.ω_{k})(ω_{1}...ω_{k})...(ω_{1}.}_{[m/k]} ..ω_{k})...].

Therefore when k ≤ [m/2], the cylinder is completely determined by its first k symbols. Put C = [ω_{1}...ω_{k}]; we say that a cylinder of length k is admissible (admis) when it is the beginning of a cylinder of L_{m} with N_{m} = k. Then we can bound (1) by

(1) ≤ ∑_{k=1}^{[m/2]} ∑_{C admis} λ(C ∩ T^{−k}C ∩ ⋯ ∩ T^{−[m/k−1]k}C)

≤ ∑_{k=1}^{[m/2]} ∑_{C admis} (sup_{C} P^{k}1I_{C})^{[m/k]−1} λ(C)

≤ ∑_{k=1}^{[m/2]} sup_{C admis} (sup_{C} P^{k}1I_{C})^{[m/k]−1} .

We first remark that T^{k} being injective over C ∈ ξ_{k}, we have

P^{k}1I_{C} ≤ 1/ inf_{A_{0}} DT^{k} ≤ 1/2.

We split the last sum in three pieces by fixing k_{0} as the biggest integer for which k_{0}^{1+ \frac{1}{α}} ≥ e^{B} where B is the constant in Lemma 3.4. We then have by using Lemma 3.4

(1) ≤ ∑_{k=1}^{k_{0}}(1/2)^{[m/k]−1} + ∑_{k=k_{0}}^{m/3}(Bk^{−1−1/α})^{m/k−2} + ∑_{m/3}^{[m/2]} Bk^{−1−1/α}.

The first and the last sum are easily shown to be summable with respect to m. For the second term, we observe that the terms (Bk^{−1−1/α})^{m/k−2} are increasing in k when k is bigger than k_{0}. A direct estimation of the sum is B3^{1/α}m^{−1/α} which is summable with respect to m.

(2): In this case, the cylinder ξ_{m}(z) has the form

ξ_{m}(z) = [\underbrace{ω_{1}...ω_{m−k}}_{m−k} \underbrace{ω_{m−k+1}...ω_{k}}_{2k−m} \underbrace{ω_{1}...ω_{m−k}}_{m−k}].

As before, we set C = [ω_{1}...ω_{m−k}], and we say that C is admissible (admis) when it is the beginning of a cylinder of L_{m} with N_{m} = k.

(2) ≤ ∑_{k=[m/2]+1}^{δm} ∑_{C admis} λ(C ∩ T^{−k}C)

≤ ∑_{k=[m/2]+1}^{δm} sup_{C admis} sup_{C} P^{k}1I_{C}.

Let first p = p(C) ≥ m − k be such that C ∈ ξ_{p} is maximal (i.e. p(C) is the smallest p for which C ∈ ξ_{p}). When p < k, since 1 ∈ C_{∗}(a) the inequality (12) and Lemma 3.4 give

sup_{C} P^{k}1I_{C} ≤ sup P^{p}1I_{C} sup_{C} P^{k−p}1 ≤ a2^{α}Bp^{−1−1/α} ≤ a2^{α}B(m − k)^{−1−1/α}.

When p ≥ k, C ∈ ξ_{k} and T^{−k}C ∩ C ≠ ∅ we have

P^{k}1I_{C} ≤ Bk^{−1−1/α}.

But k ≥ m − k ≥ (1 − δ)m, and then the sum (15) is summable for any δ < 1.

We are now ready to state and prove the main theorems of this section Theorem 3.8. For μ-almost every z ∈ X and β < β̄(α),

sup_{t≥0} |μ_{ξ_{m}(z)} (τ_{ξ_{m}(z)} > \frac{t}{μ(ξ_{m}(z))}) − exp(−t)| = O(μ(ξ_{m}(z))^{β}).

where the critical exponent β̄(α) = 1 − α.

Proof. Let ε be a positive number. Let z be a typical point for Proposition 3.7 and for the Shannon-McMillan-Breiman theorem. We want to apply Lemma 2.4; Let m(ε) such that for any m > m(ε) we have (1 − ε)m ≤ τ(ξ_{m}(z)), μ(ξ_{m}(z)) ≤ exp(−m2h_{μ}/3) and also μ(ξ_{εm}(T^{[(1−ε)m]}z)) ≥ exp(−(2[εm])h_{μ}).

For the sake of simplicity, we put for any m, U_{m} = ξ_{m}(z). For any m > m(ε) such that U_{m} is maximal, we have (1 − ε)m ≤ τ(U_{m}) ≤ m, and all the iterates T^{j}U_{m} for 1 ≤ j < m are at a distance bigger than a_{m} from the neutral fixed point (because U_{m} is maximal). If τ(U_{m}) < m then the density stays bounded on the orbit T^{j}U_{m} by ba_{m}^{−α} so we have

λ(T^{τ(U_{m})}U_{m}) ≥ \frac{a_{m}^{α}}{b} μ(T^{τ(U_{m})}U_{m}) ≥ \frac{a_{m}^{α}}{b} exp(−2εmh_{μ}).

On the other hand, when τ(U_{m}) = m we still get

λ(T^{τ(U_{m})}U_{m}) = 1 ≥ \frac{a_{m}^{α}}{b} exp(−2εmh_{μ}).

Lemma 3.5 gives us the following estimation with N = μ(U_{m})^{−α+ε},

a_{N}(U_{m}) = O(μ(U_{m})^{1−α−3ε}).

Lemma 3.1 with R = U_{m} gives us

b_{N}(U_{m}) = O_{L}((μ(U_{m})^{−α+ε} − m)^{1− \frac{1}{α}}) = O_{L}(μ(U_{m})^{(−α+ε)(1− \frac{1}{α})}).

We can then apply Lemma 2.4, which gives

c(U_{m}) =≤ a_{N}(U_{m}) + b_{N}(U_{m}) = O(μ(U_{m})^{β})

for β ≤ 1 − α − 3ε and β ≤ 1 − α − 2ε(1/α − 1). We finally end up with

d(U_{m}) = O(μ(U_{m})^{β})  (16)

for any β < 1 − α, since ε is arbitrary small, which conclude the proof by applying Theorem 2.1.

Remark 3.9. The preceding theorem shows that the critical exponent β̄(α) is smaller than 1.We point out that, by using Proposition 2.3 the power β̄ cannot exceed 1.

Theorem 3.10. For μ-almost every z ∈ X, we have for any t ≥ 0 and K ≥ 0 and β < β̄(α)

|μ_{ξ_{m}(z)} (N_{ξ_{m}(z)}(t) = K) − \frac{t^{K}}{K!} exp(−t)| = O(μ(ξ_{m}(z))^{β/(K+1)}).

with the critical exponent β̄(α) = 1 − α.

Proof. Let z be a typical point satisfying the preceding theorem and m such that U_{m} = ξ_{m}(z) is maximal.

By invoking the footnote of Theorem 2.6, it will be sufficient to use the weakened α-mixing condition

α′(M) = O_{L}((M − m)^{α− \frac{1}{α}})

given by Lemma 3.1 to apply Theorem 2.6. Take M = μ(U_{m})^{−α}; we thus find for β < 1 − α, and by the estimation (16) and Theorem 2.6 an error of the order

f(K, U_{m}) = const[d(U_{m}) + α′(M) + Mμ(U)] = O(μ(U_{m})^{β}).

By applying Theorem 2.8, the error for the probability to have K successives visits is of the order μ(U_{m})^{β/(K+1)} for all β < 1 − α.

4 Concluding remarks

We conclude with few observations. First, the proofs for the exponentialone law and the Poisson law given in section 3 for a class of non uniform hyperbolic maps, can be easily adapted, and they are even easier, to all the cases quoted in the introduction, namely: Axiom A diffeomorphisms, transitive Markov chains, expanding maps of the interval with a spectral gap and in general to all φ-mixing dynamical systems.

For such systems, an estimation for the error can also be done: following the arguments of Theorems 3.8 and 3.10, one can easily see that the critical exponent β̄ is equal to 1. This supports our believes that: (i) the error terms of type μ(U)^{β} could be optimal and (ii) the non uniform hyperbolicity of the map reflects in the critical exponent: in that case, in fact, it should be strictly smaller than one.

Acknowledgments

We would like to thank Viviane Baladi for a careful reading of a preliminary version of this work and Bernard Schmitt for useful discussions. B.S. acknowledges the ESF for support during the workshop “Probabilistic methods in non-hyperbolic dynamics” in Warwick.

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