Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors.


Introduction
Poincaré's recurrence theorem [22] established that for measure preserving maps points return to neighbourhoods arbitrarily often almost surely. The return time for the first return was quantified by Kac [17] in 1947 and since then there have been efforts to describe the return statistics for shrinking neighbourhoods. One looks at returns for orbit segments whose length is given by a parameter t scaled by the size of the target set. The scaling factor is suggested by Kac's theorem. In [19,18] it was shown that for ergodic maps one can achieve any limiting statistics if one chooses the shrinking target sets suitably. For a generating partition the natural neighbourhoods are cylinder sets and for those the limiting distributions for entry and return times were shown under various mixing conditions to be exponential with parameter 1 (see for instance [15,11,5,1]). However Kupsa constructed an example which has a limiting hitting time distribution almost everywhere and which is not the exponential distribution with parameter 1. For multiple returns it has been established under various mixing conditions that the limiting distribution is Poissonian almost surely. The first such result is due to Doeblin [9] for the Gauss map for which he showed that at the origin multiple returns to cylinder sets are in the limit Poisson distributed. Pitskel [21] (see also [7]) used the moment method to prove that the return times are in the limit Poissonian for equilibrium states for Axiom A maps using Markov partitions. Similar results have been shown for some non-uniformly expanding maps [16,10], for rational maps [12] and for φ-mixing maps [3,2]. Using the Chen-Stein method, it was also shown for toral automorphism (by way of harmonic analysis) [8] and for unions of cylinders and φ-mixing measures in [14]. More results arementioned in the review [13].
For the return times to metric balls B ρ on manifolds [6] proves the limiting distribution to be Poissonian for the SRB measure on a one-dimensional attractor which allows the construction of a Young tower [24,25] with exponentially decaying correlations. The speed of convergence turns out to be a (positive) power of the radius ρ of the target ball.
Here we prove a similar result in the case when the correlations decay at a polynomial rate. Also the attractor is not required to be one-dimensional. In fact, the result here applies to any invariant measure that can be constructed using the tower construction whether it be absolutely continuous or not. The speed of convergence in this case is a negative power of | log ρ |. Let us note that recently Pène and Saussol [20] have obtain the limiting Poisson distribution for return times for the SRB measure assuming some geometric regularity.
The results of this paper are taken from [23] and are organised as follows: In Section 2 we state the results where the exact conditions are given in later sections (indicated within the theorems). In Section 3 we give the description of Young's tower construction which is central to the result. In Section 4 we prove the main result Theorem 1 which proceeds in several steps which are outlined at the beginning of the section. In Section 5 we consider the case of a diffeomorphism on a manifold and show that very short returns i.e. those that are of order | log ρ | (where ρ is the radius of the target ball) constitute a very small portion of the manifold whose measure can be bounded. We use and adapt an argument from [6] Lemma 4.1.

Results
Let (M, T ) be a dynamical system on the compact metric space M.  Suppose that the tail of the tower's return time function decays polynomially with degree λ > 4. Let µ be the SRB measure admitted by the system and let ς be its dimension. Assume that µ is geometrically and ξ-regular. Exact assumptions are in Section 4.1.
Then there exist constants κ, κ ′ ∈ (0, λ−4 2 ) and a, C, C ′ > 0 such that for ρ sufficiently small there exists a set X ρ ⊂ M with µ(X ρ ) ≤ C ′ | log ρ | −κ ′ such that for all ρ-balls with Theorem 1 establishes the limiting statistics of returns outside the set V ρ of very short returns. If M is a manifold and the map T sufficiently regular, one can say something about the size of V ρ . This is done in the following result where we obtain for smooth maps on manifolds the following limiting result that at the same time controls the size of the forbidden set.
Theorem 2. Let (M, T ) be a dynamical system satisfying the assumptions of Theorem 1 and where T is a C 2 -diffeomorphism and λ > 9. Let µ be the invariant measure. For x ∈ X ρ ∪V ρ (a) the function S t ρ,x counting the number of visits to the ball B ρ (x) satisfies (2) for ρ sufficiently small where the constants κ ∈ (0, λ−4 2 ) and C > 0 are independent of ρ and x.
Further there exists κ ′ ∈ (0, λ−9 4 ] and C ′′ > 0 so that Note that as ρ tends to zero so does the error of approximation and the measure of the excluded set. Therefore in the limit the function S is Poisson distributed on a full measure set. If the assumption "µ geometrically and ξ-regular" is replaced with "µ absolutely continuous with respect to Lebesgue measure" (the latter condition implies the former) then we obtain the following result which generalises the main theorem of [6] from exponentially decaying correlation to polynomially decaying correlations. Moreover, here we don't require the attractor to have one-dimensional unstable manifolds. Theorem 3. Let (M, T ) be a dynamical system on a compact manifold M where T : M → M is a C 2 diffeomorphism with attractor A . Suppose the system can be modeled by a Young tower whose return time function decays polynomially with degree λ > 9. Let µ be the SRB measure admitted by the system and let ς be its dimension. Exact assumptions are in Section 6.

Young Tower
We shall use the tower method which was developed by L-S Young in [24,25] to construct invariant measures and to obtain decay rates for correlations.
(I) Partition, foliation and return time function: The base Λ ⊂ M is assumed to have a suitable 'reference measure'm. There exists a collection of pairwise disjoint subsets {Λ i } i∈N so that the following are satisfied: a) Modulo sets ofm-measure zero Λ = i Λ i . b) The base Λ is equipped with the product structure and each of the Λ i is assumed to be a rectangle (see below) and satisfy the Markov property under the return transform (see [24]). c) There exists a return time function R : Λ → N such that T R Λ i = Λ. Now, let γ u (x) be the local unstable leaf through x and γ s (y) be the local stable leaf through y. Λ i being a rectangle means that if x, y ∈ Λ i then there is a unique intersection z = γ u (x) ∩ γ s (y) which also lies in Λ i . The map T contracts along the stable leaves and similarly T −1 contracts along the unstable leaves (see Assumption (A1)). Note that, if γ u ,γ u are two unstable leaves then the homotopy map Θ : The return time function R is constant on each of the subsets Λ i , meaning that for each i there is an R i ∈ Z + such that R| Λ i = R i and in fact T R i Λ i = Λ. Without loss of generality we will assume that the greatest common divisor of all of the R i is equal to one. The function R is assumed to be integrable on each unstable leaf γ u with respect to a 'reference measure'm. That is for all local unstable leaves γ u , wherem γ u is the conditional measure on γ u . Precise assumptions are formulated in Assumptions (A1) and (A2).
Since any point x from Λ i ⊂ Λ will return to Λ after R i iterations we can define the return transformT : Λ → Λ piecewise as follows: By assumptionT Λ i = Λ withT | Λ i one to one and onto.
We extend the return time functionT to all of M in the following way. For a point x let j ≥ 0 be the smallest integer so that T j x ∈ Λ. Then we putT (x) = T j (x).
(II) Separation Time: Based on the flight time R and the partition of the base, we define the separation time s(x, y) = min {k ≥ 0 :T k x andT k y lie in distinct Λ i }, so that for x, y ∈ Λ i , s(x, y) = 1 + s(T x,T y) and in particular s(x, y) ≥ 1. If F ⊂ Λ then we also put s(F ) = min x,y∈F s(x, y).
We extend the separation function to points outside Λ as follows. If for points x, y there exists an integer k ≥ 0 (smallest) so that T k x, T k y ∈ Λ i for some i, then s(x, y) = s(T k x, T k y).
(III) The Jacobian: Even though the original system (M, B, T, µ) is not necessarily differentiable in the ordinary sense, one uses the Radon-Nikodym derivative of T with respect to the 'reference measure'm (following [24], p. 596). The derivative exists and is well defined because every T R i | Λ i and its inverse are non-singular with respect to the conditional measurem γ u on unstable leaves γ u . Let The requirements on JT will be spelled out in Assumption (A1).
(IV) The SRB measure: According to [24,25] (M, T ) has a generalised SRB measure µ given by where m is the generalised SRB measures for the uniformly expanding system system (Λ,T ). If we denote by m γ u the conditional measure on unstable leaves γ u , then dm = dm γ u dν(γ u ) where dν is the transversal measure. We will refer to the portions of the tower above each Λ i as beams and assume that for n ∈ N, there are only finitely many i's for which R i = n, i.e. for every n there are only finitely many beams with that height. The map T maps each level bijectively onto the next and the last level is bijectively mapped onto all of Λ. (Note that R i may not be the first time Λ i returns to Λ.) In this way m can be extended to the entire tower by

Proof of Theorem 1
In this section we prove Theorem 1. We begin by stating the precise assumptions necessary for the result and derive some of the consequences that follow with minimal work. In Section 4.3 we introduce cylinder sets. In order to approximate the metric balls we will restrict to those cylinder set that have that have only short returns. We then provide several results on the behaviour of cylinder sets under suitable applications of the return mapT . The succeeding Section 4.4 contains estimates concerning the portions of the space M which have to be omitted in order to obtain good asymptotic behaviour for the long returns. This is the 'forbidden set'. Section 4.5 utilizes the Poisson approximation theorem from Section 7 to establish a splitting of the error term to the Poisson distribution into two parts R 1 and R 2 . The remainder of the section is devoted to estimating these error terms one by one. In Section 4.6 we estimate the error R 1 which comes from long term interactions and uses decay of correlations. Sections 4.7 and 4.8 are devoted to bounding the term R 2 which comes from short time (but not very short time) interactions. This is the place where the Young tower construction comes to play and where we have to use approximations by cylinder sets in order to make careful distinctions between short returns and long returns to balance out different contributions to the error term. In Section 4.9 the different error terms are brought together and the various parameters are optimised.

4.1.
Assumptions. Let (M, T ) be a dynamical system equipped with a metric d and let µ be the SRB measure associated to the system (whose existence follows by [24,25] from Assumptions (A1) and (A2)). We will require the following: (A1) Regularity of the Jacobian and the metric on the leaves There exists a constant C 0 > 0 and α ∈ (0, 1) such that for any x, y in Λ with s(x, y) ≥ 1 (A2) Polynomial Decay of the Tail There exist constants C 1 and λ > 4 such that for every unstable leaf γ u . With regard to Assumption (A1)(d) this condition is satisfied for all unstable leaves if it can be verified for a one γ u .
(A3) Additional assumption on λ Let ς be the dimension of the measure m γ u andς the dimension of µ. We will require that so that for ρ small enough: (a) (ξ-regularity) There exists w 0 ∈ (1, ξ) and a > 0 so that We say µ is ξ-regular. (b) (geometric regularity) There exists a ς ′ < ς,ς ′′ >ς satisfying (4) and C 2 > 0 such that for which x ∈ E ρ and all unstable leaves γ u .

4.2.
Immediate Consequences of the Assumptions. In this section we list some basic results which will be needed in the proof of the main results.
There exists a constant C 3 > 1 such that (i) for any x and y in Λ, γ u (x) = γ u (y) with separation time s(x, y) ≥ q: Proof. (i) Let x, y ∈ Λ, , γ u (x) = γ u (y) and let q ≥ 1 be an integer less than or equal to s(x, y). Then by the chain rule and (A3)(a) (ii) By the Mean Value Theorem for some x ∈ F and x ′ ∈ F ′ . The result follows now from part (i).
According to [25](Theorem 3) the decay of correlations is polynomial: Let φ be a Lipschitz continuous function and ψ ∈ L ∞ constant on local stable leaves. Then one has , for some C 4 > 0 where λ > 0 is the tail decay exponent from assumption (A2). Note that in general for functions ψ which are not constant on local stable leaves, the supremum norm on the RHS of (7) has to be replaced by the Lipschitz norm (see [24]).
We will also need the following function of s ∈ R + : Lemma 4.2 (Decay of Ω). There exists a constant C 5 such that for s ≥ 4 using the tail decay, where c 1 < ∞. We complete the proof by setting C 5 = √ c 1 .

4.3.
Cylinder sets. Let s be a given integer. We shall separate the beams with return times greater than s and also portions of the base that visit those beams during the "flight". The beams with heights less than s will be referred to as "short" and constitute the principal part. The "tall" beams (i.e. when the returns are > s) will be treated like error terms and contribute to the "forbidden" set X ρ whose size is small and estimated in Section 4.4. Let us introduce several quantities that will be needed to deal with the long return times.
(I) For indices (i 0 , . . . , i l ) ∈ N l+1 we define the l-cylinder (w.r.t. the mapT ) by and denote by I the collection of indices (i 0 , . . . , i l ) such that the associated cylinder ζ i 0 ,...,i l is non-empty.
(II) For every Λ i let's define the subset consisting exclusively of points that only visit short beams:Λ Let us note that if the original beam Λ i happens to be tall (i.e. R i > s) then the correspondingΛ i will be empty.
Similarly we define the restriction of a cylinder ζ i 0 ,...,i l to short returns bỹ In other words In particular we see that if one of the beams on the cylinder's path is tall, i.e. R i j > s for a j ∈ [0, l], then it must have originated inside Λ i 0 \Λ i 0 . Thus as long as l ≤ n.
(III) For given n, j and i 0 ∈ N, j < R i 0 ≤ s we define the set of 'suitable' symbols by (10) Accordingly a string of symbols Note that for any given values j and n the cylinders indexed by I i,j,n partition the beam base Λ i , up to set of measure zero, i.e.
In what follows we shall often write I, instead of I i,j,n .
The remainder of this section is taken up by providing some essential estimates involving the quantities introduced.
Lemma 4.3. The diameter of the cylinder set ζ i 0 ,...,i l restricted to unstable leaves is exponentially small: Proof. Let x and y be two points in ζ i 0 ,...,i l , then by definition we havê It follows that s(x, y) ≥ l + 1. Therefore by Assumption (A1b): and, since the points x, y were arbitrary, we conclude that for any set B ⊂ T −n B ρ and for allζ τ ′ = ∅.
Proof. By inclusion we have Let the number b be such that n + j − b = R l . Recall that τ ∈ I means that it is (n, j)-minimal, i.e. that n + j lies between R l and R l+1 , Recall that s(ζ τ ′ ) = l, thus we can push both the numerator and the denominator forward byT l and use distortion, Lemma 4.1 and (A1)(a), to obtain for ζ τ ′ = ∅: since by assumption and (8) For the numerator we obtain for some c 1 , and for the denominator we use that µγu(Λ) ≥ c 2 for some c 2 > 0. The lemma now follows with Then there exists a constant C 7 such that where we use the notation γ s (B) = γ s :γ s ∩B γ s .
Proof. All the unions, sums and maxima in this proof are subscripted with "τ ′ | ∃τ ⊂ τ ′ , τ ∈ I i 0 ,j,n , T −j B ρ ∩ ζ τ ′ = ∅," unless otherwise specified. From Lemma 4.3 we know that for unstable leaves γ u . Without loss of generality we can assume thatζ τ ′ = ζ τ ′ . Since ζ τ ′ contains a cylinder τ satisfying (n, j)-minimality, we deduce from n + j < l k=0 R i k ≤ (l + 1) s a lower bound on l: Thus and for j < R i 0 we can further say by Assumption (A1)(b) that As the above estimate works for any cylinder whose subscript τ ′ satisfies "τ ′ | ∃τ ⊂ τ ′ , τ ∈ and we can conclude

4.4.
Measure of the forbidden set X ρ . We will need the following lemma. For ω ∈ (0, 1) and ρ ∈ (0, 1) define There exists an integer p(D) such that We will also need the following result on the size of the set where tall towers dominate, that is where R i is larger than s. Recall that Ω(s) Lemma 4.7. For n, s ≥ 1 there exist sets D n,s ⊂ M such that the non-principal part contributions are estimated as for any B ⊂ B ρ (x) and x ∈ D n,s where (p(D) as above) Proof. We employ Lemma 4.6, with ℓ 0 = µ and ℓ 1 (·) = i from Lemma 4.6 we know that µ(D n,s ) ≤ p(D) ( √ n + 2 Ω(s)) −1 ℓ 1 (M). Since R(T l x) > s exactly if x ∈T −l {R > s} we get SinceΛ i = ∅ for R i > i we bound the measure ℓ 1 (M) as follows Outside the set D n,s we have Let η ∈ ( 3 λ−1 , 1 2 ) and β < 1− 3 λ−1 be according to Assumption (A4). Putσ = λ−1 2 min {η, (1 − β)}− 3 2 . By assumptionσ > 0 and we can finally estimate the size of the forbidden set defined by X ρ = p−1 n=J D n,n η ∪ D n,n 1−β ∪ E ρ whose parts have essentially been estimated in the previous lemma and assumption (A4).
Proposition 4.1. There exist a constant C 8 such that Proof. We estimate the contributions to X ρ separately in the three following paragraphs.
(I) By Lemma 4.7 and since Ω(s) as η + 1 − ηλ < 0 and J = ⌊a | log ρ |⌋. The term with s = n 1−β is estimated similarly. Therefore Combining the estimates from (I) and (II) results in Poisson approximation of the return times distribution. To prove Theorem 1 we will employ the Poisson approximation theorem from Section 7. Let x be a point in the phase space and B ρ : where t is a positive parameter. We write S b a = b n=a X n (and S = S N 1 ). Then for any 2 ≤ p ≤ N (C 12 from Section 7) Since we restrict to the complement of the set V ρ (cf. (1)) we have from now on where J = ⌊ a | log ρ | ⌋. Note that if k > N then P(S = k) = 0 and (14) using the fact that µ(B ρ ) ρ ς ′ and for ρ sufficiently small.
We now proceed to estimate the error between the distribution of S and a Poissonian for k ≤ N based on Theorem 7.1.

4.6.
Estimating R 1 . By invariance of the measure µ we can also write We now use the decay of correlations (7) to obtain an estimate for R 1 . Approximate ½ Bρ by Lipschitz functions from above and below as follows: φ(x) = 1 on B ρ 0 outside B ρ+δρ andφ(x) = 1 on B ρ−δρ 0 outside B ρ with both functions linear within the annuli. The Lipschitz norms of both φ andφ are equal to 1/δρ andφ ≤ ½ Bρ ≤ φ.
We obtain The two terms X and Y are estimated separately. The first term is estimated as follows: In order to estimate the second term Y we use the decay of correlations and have to approximate ½ S N−j−p For integers n and s letΛ i be as before, then by Lemma 4.7 for x ∈ D n,s we have where B = B ρ+α l \ B ρ−α l . For points y ∈ Λ \Λ we let l be so that R l (y) ≤ n < R l+1 (y) then we get l ≥ n/s ≥ n β where we choose s(n) = n 1−β and β < 1 so that ∞ n g(n β ) −1 < ∞ in accordance with Assumption (A4). By the contraction property diam(T n γ s (y)) ≤ α n/s ≤ α n β for all y ∈ Λ \Λ. Consequently where we used w(n) = n β log α log ρ . If we split p = p ′ + p ′′ then we can estimate as follows: where the triple sum on the RHS is by Lemma 4.7 bounded by where in the last estimate we put p ′ = p/2. Now let δρ = ρ w where w ∈ (1, ξ) is chosen in accordance with Assumption (A4) (this is possible since ξ = ς(λ − 2) > 1). Hence In the same way we obtain a lower estimate. Since D n,n 1−β ⊂ X ρ for n = J, J + 1, . . . we conclude that for x ∈ X ρ one has:

4.7.
Estimating the individual terms of R 2 (for n fixed). We will estimate the measure of each of the summands comprising R 2 individually with the help of the Young tower. Fix n and for the sake of simplicity we will denote B ρ ∩ T −n B ρ by B n . Then With s = ⌊ n η ⌋ letΛ i be as in Section 4.3 (II), then using Lemma 4.7 for the second term on the RHS to the complement of the set D n,s . Since (ζ as in Section 4.3) we get, since by (9) iΛ i ⊂ τ ∈Iζ τ , for each of the summands in the principal term on the RHS of (17) where we used Lemma 4.4 in the last step. From Lemma 4.5 we obtain a bound for the sum of measures of the cylinders, whence using the product structure of the measure m. Note that since B ρ+C 7 α n/s (x) ⊂ B 2ρ (x) ∪ B 2C 7 α n/s (x) we obtain on unstable leaves by Assumption (A1)(b) that using the geometric regularity (A4)(b) provided that the radius 2C 7 α n/s is small enough. Since n ≥ J and s depends on n we can guarantee the above radius to be sufficiently small provided that ρ is small enough. Therefore for some c 1 . Thus the first term (principal term) on the RHS of (17) can be bounded as follows for a constant C ′ 6 .
4.8. Estimating R 2 . Combining inequalities (17) and (18) results in provided that n ≥ J and the center x of the ball B ρ lies outside the set D n,s . As before let s = ⌊n η ⌋ where η ∈ ( 3 λ−1 , 1). Summing up the B n terms over n = J, . . . , p − 1, we see that outside the set of forbidden ball centers V ρ ∪ X ρ we get with For ρ small and by Lemma 4.2 Ω(s) ≤ C 5 s −θ where θ = λ−1 2 we obtain , for ρ small, we get for some C 9 . Note that the above it true provided ρ is sufficiently small and the center x of the ball B ρ is not in X ρ ∪ V ρ .

4.9.
Estimate of the total error. Now we want to bound all of the error components from inequality (13) with terms of the order of J −σ or a negative power of | log ρ |, where σ = η λ−1 2 − 3 2 < λ−4 2 . To that end we choose the length of the gap p to be and estimate the three error terms on the RHS of (13) separately.
(I) The last summand is immediately estimated as (II) For the term involving R 1 we obtain Let w < ξ and ς ′ < ς,ς ′′ >ς so that ς ′ (λ − 1) −ς ′′ − w > 0. Since µ(B ρ ) ≥ ρς ′′ for ρ small enough we get for the first term on the RHS for ρ small: and so (with some c 5 , c 6 ) since by Assumption (A4)(a) 3 2 (III) Utilizing the estimate from Section 4.8 and using the fact that N = ⌊t/µ(B ρ )⌋ yield Combining the results of estimates (I), (II) and (III) above we obtain for ρ sufficiently small the RHS of (13) as follows (x ∈ X ρ ) for some c 8 , where κ = min{σ, a} is positive as long as v < σ. This now concludes the proof of Theorem 1 as it shows that for ρ small enough and for any x / ∈ X ρ we have for By (14) this estimate also extends to k > N. By Proposition 4.1 the size of the forbidden set is then µ(X ρ ) = O(| log ρ | −κ ′ ) where κ ′ =σ. The proof of Theorem 1 is thus complete.

Very Short Returns and Proof of Theorem 2
In this section we prove Theorem 2 which is a direct consequence of Theorem 1 and Proposition 5.1 below, which estimates the measure of the set with short return times. We first state the precise assumptions. Again we use the Young tower construction. 5.1. Assumptions. Let (M, T ) be a dynamical system equipped with a metric d. Assume that the map T : M → M is a C 2 -diffeomorphism. As at the start of the paper the set V ρ ⊂ M is given by where J = ⌊ a | log ρ | ⌋ and a = (4 log A) −1 with ). Suppose the system can be modeled by a Young tower possessing a reference measure m and that the greatest common divisor of the return times R i is equal to one. Let µ be the SRB measure associated to the system. We require the following in Proposition 5.1: (B1) Regularity of the Jacobian and the metric on the Tower The same as Assumption (A1) (B2) Polynomial Decay of the Tail There exist constants C 1 and λ > 9 such that (B3) Geometric regularity of the measure Let ς be the dimension of the measure µ. Suppose that the positive constant ς ′ < ς is fixed. There exists a constant C 2 > 0 and a set E ρ ⊂ M satisfying µ(E ρ ) ≤ C 2 | log ρ | −λ/2 such that m γ u (B ρ (x)) ≤ C 2 ρ ς ′ for all x ∈ E ρ and ρ small. Note that in Proposition 5.1 we don't require the measure to be ξ-regular.

5.2.
Estimate on the measure of V ρ . Before we prove the main result of this section we shall present a lemma which will be needed in the proof of Proposition 5.1.

Proof. By assumption
Since ρ 1 2 = e − 1 2 | log ρ | the statement of the lemma follows for ρ small enough.
Now we can show that the set of centres where small balls have very short returns is small. To be precise we have the following result: Proposition 5.1. There exist constants C 10 > 0 such that for all ρ small enough Proof. We largely follow the proof of Lemma 4.1 of [6]. Let us note that since T is a diffeomorphism one has We partition V ρ into level sets N ρ (n) as follows The above union is split into two collections V 1 ρ and V 2 ρ , where aand where the constant b ∈ (0, 1) will be chosen below. In order to find the measure of the total set we will estimate the measures of the two parts separately.
(I) Estimate of V 2 ρ We will derive a uniform estimate for the measure of the level sets N ρ (n) when n > bJ as µ(N ρ (n)) = µ(T −n N ρ (n)) and where here we put s = J 1 4 which means On the RHS the first term is the principal term, and the other two terms will be treated as error terms. Let us first estimate the error term. Note that like in the proof of Lemma 4.7 and assumption (B2) the second sum on the RHS can be bounded by . As for the first sum (principal term) on the RHS in (20), we can decompose the beam baseΛ i into cylinder sets, as in Section 4.3, with the index set I defined just as in (10) Incorporating the above estimates and decomposition into (20), we obtain (c 2 = c 1 + C 2 5 ) We will consider each of the measures m(T −(n+j) N ρ (n)∩ζ τ ′ ) separately. Let τ = (i 0 , . . . , i l ) ∈ I i 0 ,j,n and τ ′ = (i 0 , . . . , i l−1 ) as in Section 4.3. By distortion of the Jacobian, Lemma 4.1, we obtain forζ τ ′ = ∅ (which impliesζ τ ′ = ζ τ ′ ): where, as before,γ u = γ u (T l x) for x ∈ ζ τ ′ ∩ γ u . For the last line compare with (12). We estimate the numerator by finding a bound for the diameter of the set. Let the points x and z in T −n N ρ (n) be such that T −j x, T −j z ∈ T −(n+j) N ρ (n) ∩ζ τ ′ ∩ γ u for an unstable leaf γ u . From the from (n, j)-minimality of τ (R l ≤ n + j < R l+1 ) we know that Note that T n x, T n z ∈ N ρ (n) and so d(T n x, T n z) ≤ d(T n x, x) + d(x, z) + d(z, T n z) ≤ 4A n ρ + d(x, z).
Then employing (A1)(b) and keeping in mind that n > bJ Taking the supremum over all points x and z yields By assumption (B3) on the relationship between the measure and the metric for ς ′ < ς, which implies by the product structure of m that Incorporating the estimate into (22) yields where c 4 ≤ 2c 2 C 3 m(Λ) . Substituting this into estimate (21) we see that (n ≤ J) Next we have to bound the triple sum on the RHS. As we showed before all of theζ τ ′ with τ ′ | ∃τ ∈ I are disjoint and are all subsets ofΛ i , therefore Hence for n = ⌈ bJ ⌉ , . . . , J we obtain for some c 5 and ρ small enough, and consequently (23) µ for some constant c 6 (and ρ small enough) as J = ⌊a| log ρ |⌋.
Here we consider the case 1 ≤ n ≤ ⌊ bJ ⌋. Following [6] we put By [6] Lemma B.3 one has N ρ (n) ⊂ N spρ (2 p n) for any p ≥ 1, and in particular for p(n) = ⌊ lg bJ − lg n ⌋ + 1. Therefore Now define n ′ = n2 p(n) and ρ ′ = s p(n) ρ. A direct computation shows that 1 ≤ n ≤ ⌊ bJ ⌋ implies ⌈ bJ ⌉ ≤ n ′ ≤ 2bJ and so Therefore to estimate the measure of V 1 ρ it suffices to find a bound for N ρ ′ (n ′ ) when n ′ ≥ bJ. This is accomplished by using an argument analogous to the first part of the proof. We replace all the n with n ′ and ρ with ρ ′ . The cutoff J 1/4 = (⌊ a | log ρ | ⌋) 1/4 remains unchanged. We get for b < 1/3 and thus obtain an estimate similar to (23):

(III) Final estimate
Overall we obtain for all ρ sufficiently small where C 10 = c 6 + c 8 .

Recurrence under an Absolutely Continuous Measure
Here we consider measures that are absolutely continuous with respect to Lebesgue measure.
Let d be the metric on M and T : M → M a C 2 -diffeomorphism with attractor A . We assume that the system can be modeled by a Young tower possessing a reference measurê m, and that the greatest common divisor of the return times R i is equal to one. Let µ be the SRB measure on the attractor, that is µ γ u is absolutely continuous with respect to Lebesgue ℓ on the unstable leaves γ u where its density function f is regular and bounded. µ γ u (F ) = F f dℓ for F on an unstable leaf γ u . By [25] 1 c ≤ f (x) ≤ c for a.e. x ∈ A and some c > 1. We require Assumptions (A1), (A2) and (A3) to be satisfied with λ larger than 9. Note that the attractor A ⊂ M is given by 6.1. Regularity of the SRB measure µ. Let us note that by [6] Lemma B.2 there exists a ς > 0 and a set U ′ ρ ⊂ M such that µ(U ′ ρ ) ≤ c 1 Ω(| log ρ |) for some c 1 and so that µ(B ρ (x)) ≤ ρ ς for all ρ ∈ (0, 1] and all x ∈ U ′ ρ . Hence µ is geometrically regular for ς. The following proposition shows that the SRB measure µ is ξ-regular for any ξ > for every ρ > 0 and for every x ∈ U ′′ ρ . Proof. Put A = B ρ+ρ w (x) \ B ρ (x). For s, l > 0 we let as in Section 4.3Λ i = {x ∈ Λ i : R(T k ) ≤ s ∀k ≤ l}. Then as in (17) we use Lemma 4.7 to obtain for x ∈ D sl,s : where |ζ τ ′ | ≤ α l . Here we proceeded as in (18) and putγ u = γ u (T l x) for some x ∈ ζ τ ′ ∩ γ u . As before, the second term on the RHS estimates the contributions from terms m(T −j A ∩ (Λ i \Λ i )) and from the tall beams where R i > s. AsT l is one-to-one onζ τ ′ we get where u is the dimension of the unstable manifolds and the diameter is measures inside the unstable leaf. Here we used that m on the unstable manifolds (of dimension u) is absolutely continuous with respect to the Lebesgue measure and the fact that the map T expands distances on M by at most a factor A at each iteration. Since the map T is C 2 on M we get thatζ τ ′ ∩ γ u are on nearly flat segment of an unstable leaf γ u , which allows us to estimate the measure of the intersection A ∩ζ τ ′ ∩ γ u . Since A is an annulus of thickness ρ w there exists a constant c 2 such that m γ u (A ∩ζ τ ′ ) ≤ c 2 ρ w+1 2 . SinceT lζ τ ′ = Λ ifζ τ ′ = ∅ (see Section 4.3 (II)) we obtain (for ς ′ < u) Let w 0 = 2 u (D + 1) − 1 and k(w) = w−w 0 2 log A for w > w 0 . Then put s = k(w)| log ρ |η and l = | log ρ | 1−η , whereη ∈ (0, 1). Thus 1 2 | log ρ | ≤ sl ≤ 2| log ρ | and we obtain (A sl ρ w+1 2 ) ς ′ ≤ ρ D+1 provided w > w 0 and ς ′ < u close enough to u. We have ρ D+1 ≤ µ(B ρ (x)) for all x ∈Û ′′ ρ whereÛ ′′ ρ ⊂ M is a small set whose measure is by [6] Lemma A.1 bounded by O(ρ) for some constant C 11 . This implies µ(A) ≤ c 5 µ(B ρ ) ρ ς ′ + α ς ′ l + √ sl Ω(s) ≤ c 6 µ(B ρ ) ρ ς ′ + α ς ′ | log ρ | 1−η + (k| log ρ |) −η λ−1 for a constant C 11 and for all ρ small enough. Since we can chooseη arbitrarily close to 1, we obtain any exponent a < λ−2 2 . This applies for points x ∈ U ′′ ρ , where the measure of the forbidden set U ′′ ρ =Û ′′ ρ ∪ D sl,s is bounded by O(1)((w − w 0 )| log ρ |) −a . 6.2. Proof of Theorem 3. To prove this result we combine Theorems 1 and 2. Let J = ⌊ a | log ρ | ⌋ as before, and take a = [4 log( DT L ∞ + DT −1 L ∞ )] −1 . Clearly the dimension ς of the measure m γ u is equal to the dimension u.
With the forbidden set Z ρ = X ρ ∪ V ρ ∪ E ρ whose size is bounded by we get that for any point x ∈ A \ Z ρ the function S counting the number of visits to the ball B ρ (x) can be approximated by a Poissonian, that is P(S = k) − e −t t k k! ≤ C | log ρ | −κ for all k ∈ N 0 , for any κ < λ−7 4 by the remark following the proof of Theorem 1. Now we setĈ = max(C, C ′′ ).

Poisson Approximation Theorem
This section contains the abstract Poisson approximation theorem which establishes the distance between sums of {0, 1}-valued dependent random variables X n and a random variable that is Poisson distributed. It is used in Section 4.5 in the proof of Theorem 1 and compares the number of occurrences in a finite time interval with the number of occurrences in the same interval for a Bernoulli process {X n : n}.
Proof. Let (X n ) n∈N be a sequence of independent, identically distributed random variables taking values in {0, 1}, constructed so that P(X 1 = 1) = ǫ. Further assume that theX n 's are independent of the X n 's. LetS = N n=1X n . Then Thanks to [4] we can bound the second sum using the estimate For summands of the remaining term we utilize the proof of Theorem 2.1 from [6] according to which for every k ≤ N, | P(S = k) − P(S = k) | ≤ 2N(R 1 + R 2 + pǫ 2 ) + 4pǫ.