Quantitative recurrence of some dynamical systems with an infinite measure in dimension one

We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of $\mathbb{Z}$-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.


Introduction
The quantitative recurrence properties of dynamical systems preserving a probability measure have been studied by many authors since the work of Hirata [6]. Some properties are defined by estimating the first return time of a dynamical system into a small neighbourhood of its starting point. Results in this concern have been described in [14], let us mention works in this situation [1,15]. This question has been less investigated in the context of dynamical systems preserving an infinite measure. In [3], Bressaud and Zweimüller have established first results of quantitative recurrence for piecewise affine maps of the interval with infinite measure. The case of Z 2 -extension of mixing subshifts of finite type has been investigated in [11]. Results have been also established for random walks on the line [12], for billiards in the plane [10] and for null-recurrent Markov maps in [13]. A measure-preserving dynamical system is given by (X, B, µ, T ) where (X, B) is a measurable set, µ is a finite or σ-finite positive measure and T : X → X is a measurable transformation preserving the measure µ (i.e. µ(T −1 A) = µ(A), for every A ∈ B). We are interested in the case where µ is σ-finite. We assume that X is endowed with some metric d X and that B contains the open balls B(x, r) of X. Our interest is in the first time the orbit comes back close to its initial position. For every y ∈ X, we define the first return time τ ǫ of the orbit of y in the ball B(y, ǫ) as: τ ǫ (y) := inf{n ≥ 1 : T n (y) ∈ B(y, ǫ)} ∈ N ∪ {+∞}.
We consider conservative dynamical systems, that is dynamical systems for which the conclusion of the poincaré theorem is satisfied. This ensures that, for every ǫ > 0, τ ǫ < ∞, µ almost everywhere. The main goal of this article is to study the behavior of τ ǫ as ǫ → 0. A classical example of dynamical systems preserving an infinite measure is given by Z-extensions of a probability-preseving dynamical system. Given a probability-preserving dynamical system (X,B, ν,T ) and a measurable function ϕ :X → Z, we construct the Z-extension (X, B, µ, T ) of (X,B, ν,T ) by setting X :=X × Z, B :=B ⊗ P(Z), µ := ν ⊗ l∈Z δ l and T (x, l) = (T (x), l + ϕ(x)). We endow X with the product metric given by d X ((x, l), (x ′ , l ′ )) := max{dX (x, x ′ ), | l − l ′ |}. Hence T n (x, l) = (T n x, l + S n ϕ(x)), where S n ϕ is the ergodic sum S n ϕ := n−1 k=0 ϕ •T k . Therefore, for ǫ small enough, T n (x, l) ∈ B((x, l), ǫ) ⇐⇒T n (x) ∈ BX (x, ǫ) and S n ϕ(x) = 0.
Our main results concern the case when (X,B, ν,T ) is a mixing subshift of finite type (see Section 3 for precise definition), which are classical dynamical systems used to model a wide class of dynamical systems such as geodesic flows in negative curvature, etc. Consider (X,B, ν,T ) a mixing subshift of finite type and ν a Gibbs measure associated to a Hölder continuous potential. Moreover we have a ν-centered Hölder continuous function ϕ. Then we get (1.1) lim ǫ→0 log τ ǫ log ǫ = −2d, µ-almost everywhere, where d is the Hausdorff dimension of ν. Moreover the following convergence holds in distribution with respect to any probability measure absolutely continuous with respect to µ: where E and N are two independent random variables with respective exponential distribution of mean 1 and standard normal distribution (see Theorem 2.1 and Theorem 2.2 for precise statements). Roughly speaking the strategy of our proof is that there is a large scale (corresponding to S n ϕ(x)) and a small scale (corresponding toT n (x)), which behave independently assymptotically. To enlight this strategy, we start out this paper with the study of the toy probabilistic model (Y n , S n ), where (S n ) n is the simple symmetric random walk and (Y n ) n is a sequence of independent random variables, with uniform distribution on (0, 1) d and where S n and Y n are independent. For this simple model, we obtain the same results. More precisely, we prove that (1.1) holds almost surely and that (1.2) holds in distribution.

toy probabilistic model
Let d ∈ N. In this section, we give a real random walk (M n ) n≥0 with values in R×]0, 2.1. Description of the model and statement of the results. The random process M n is given by M n = (S n , 0) + Y n . (S n ) n≥0 and (Y n ) n≥0 are independent such that: • Y n is uniformly distributed on (0, 1) d .
• S n is the simple symmetric random walk on Z given by S 0 = 0, i.e. S n = n k=1 X k , where (X k ) k is a sequence of independent random variables such that: P(X k = 1) = P(X k = −1) = 1/2.
We want to study the asymptotic behavior, as ǫ goes to 0, of τ ǫ for the metric associated to some norm on R d . Let c be the Lebesgue measure of the unit ball in R d . We will prove the following: Theorem 2.1. Almost surely, log τǫ − log ǫ converges to 2d as ǫ goes to 0.
For this constant c > 0, we have the following result: where E and N are two independent random variables, E having an exponential distribution of mean 1 and N having a standard Gaussian distribution.

2.2.
Proof of the pointwise convergence of the recurrence rate to the dimension. M 0 is in )0; 1( d , let ǫ so small that B(M 0 ; ǫ) is contained in )0, 1( d . Note that Leb(B(x, ǫ)) = cǫ d . We define for any p ≥ 0 the p th return time R p of (M n ) n in )0; 1( d , setting R 0 = 0, by induction : We have the relation: We will study the asymptotic behavior of the random variables R n and T ǫ and use the relation between them to prove Theorem 2.1.

2.2.1.
Study the return of the random variable R n . Proposition 2.3 (Feller [4]). There exists C > 0 such that: Due to the strong Markov property, the delays U p := R p − R p−1 between successive return times are independent and identically distributed.
Lemma 2.5. Almost surely, log √ Rn log n converges to 1 as n goes to ∞.
Proof. The proof of the lemma directly holds, once the following inequality is proved: ∀α ∈ (0, 1), ∃n 0 , ∀n ≥ n 0 , n 1−α ≤ R n ≤ n 1+α Let α ∈ (0, 1), by independence (using Remark 2.2.1), we have: Due to the asymptotic formula given in Proposition 2.1, for n sufficiently large This allows us to get the first inequality of (2.2) by using the Borel Cantelli lemma. Again, using proposition 2.1, Note that one can see, i converges almost surely to E R 1 2+2α 1 < ∞ due to the strong law of large numbers. Hence R n = O(n 2+2α ) almost surely, from which we get the second inequality.

2.2.2.
Study the return of the random variable T ǫ . In this subsection the asymptotic behavior of the random variable T ǫ is illustrated in the following lemma.
The random variable T ǫ has a geometric distribution with parameter λ ǫ := cǫ d . For any α > 0, a simple decomposition gives: The first term is handled by the Markov inequality: While the second term using the geometric distribution: Let us define ǫ n := n −2 α . Thus (ǫ n ) n≥1 is a decreasing sequence of real numbers, and T ǫ is monotone in ǫ, so that: According to Borel Cantelli lemma Proof of Theorem 2.1 The theorem follows from the two previous lemmas 2.5 and 2.6, since: Hence, we get: log τ ǫ − log ǫ → 2d as ǫ → 0 a.s.

2.3.
Proof of the convergence in distribution of the rescaled return time.
Proposition 2.7. The sequence of random variables ( Rn n 2 ) n converges in distribution to N −2 where N is a standard Gaussian random variable.
The proof of this proposition follows from the two following successive lemmas; the proof of which is straightforward and is omitted.
Lemma 2.9. The moment generating function of Gaussian random variable.
Lemma 2.10. (λ ǫ T ǫ ) ǫ converges in distribution to an exponential random variable E of mean 1.
Proof. Given Y 0 , T ǫ has a geometric distribution of parameter λ ǫ = λ(B(Y 0 , ǫ)). Let t > 0, it follows that, for E a random variable which follows exp (1), Proof of Theorem 2.2. Let us prove that the family of couples λ ǫ T ǫ , converges in distribution, as , where E and N −2 are assumed to be as above and independent. Let s > 0 and t ∈ R ,then using the independence of (T ǫ ) ǫ and (R n ) n , we get: This latter goes to 0 as ǫ goes to 0, due to Proposition 2.7. Moreover by Lemma 2.10, P(λ ǫ T ǫ > s) → P(E > s) as ǫ → 0, hence: This proves that the couple λ ǫ T ǫ , converges in distribution, as ǫ goes to 0, to (E, N −2 ).

Z-extension of a mixing subshift of finite type
Let A be a finite set, called the alphabet, and let M be a matrix indexed by A × A with 0-1 entries. We suppose that there exists a positive integer n 0 such that each component of M n0 is non zero. The subshift of finite type with alphabet A and transition matrix M is (Σ, θ), with where m is the greatest integer such that w i = w ′ i whenever |i| < m, and the shift θ : Σ → Σ, θ((w n ) n∈Z ) = (w n+1 ) n∈Z . Let ν be the Gibbs measure on Σ associated to some Hölder continuous potential h, and denote by σ 2 h the asymptotic variance of h under the measure ν. Recall that σ 2 h vanishes if and only if h is cohomologous to a constant, and in this case ν is the unique measure of maximal entropy.
For any function f : Σ → R we denote by S n f := Σ n−1 l=0 f • θ l its ergodic sum. Let us consider a Hölder continuous function ϕ : Σ → Z, such that ϕdν = 0. We consider the Z-extension F of the shift θ by ϕ. Recall that for every (w, l) ∈ Σ × Z, we have µ (B Σ×Z ((w, l), ǫ)) = ν(B Σ (w, ǫ)). We want to know the time needed for a typical orbit starting at (x, m) ∈ Σ × Z to return ǫ-close to the initial point after iterations of the map F . By the translation invariance we can assume that the orbit starts in the cell m = 0. Recall that We know that there exists a positive integer m 0 such that the function ϕ is constant on each m 0 -cylinders. Let us denote by σ 2 ϕ the asymptotic variance of ϕ: We assume that σ 2 ϕ = 0 (otherwise (S n ϕ) n is bounded). We reinforce this by the following non-arithmeticity hypothesis on ϕ: We suppose that, for any u ∈ [−π; π]\{0} the only solutions (λ, g), with λ ∈ C and g : Σ → C measurable with |g| = 1, of the functional equation is the trivial one λ = 1 and g = const. The fact that there is no non constant g satisfying (3.1) for λ = 1 ensures that ϕ is not a coboundary and so that σ 2 ϕ = 0. The fact that there exists (λ, g) satisfying (3.1) with λ = 1 would mean that the range of S n ϕ is essentially contained in a sub-lattice of Z; in this case we could just do a change of basis and apply our result to the new reduced Z-extension. We emphasize that this non-arithmeticity condition is equivalent to the fact that all the circle extensions T u defined by T u (x, t) = (θ(x), t + u.ϕ(x)) are weakly mixing for u ∈ [−π; π]\{0}. In this section we obtain the following results: Theorem 3.2. The sequence of random variables ν((B ǫ (.)) τ ǫ (.) converges in distribution with respect to every probability measure absolutely continuous with respect to ν as ǫ → 0 to E |N | , where E and N are independent random variables, E having an exponential distribution of mean 1 and N having a standard Gaussian distribution.
Corollary 3.3. If the measure ν is not the measure of maximal entropy, then the sequence of random variables converges in distribution as ǫ → 0 to a centered Gaussian random variable of variance 2σ 2 h .
3.1. Spectral theory of the transfer operator and Local Limit Theorem. In this subsection, we follow [11] to adapt our results. To begin with, let us define: the set of all one-sided infinite sequences of elements of A, endowed with the metricd((w n ) n≥0 , (w ′ n ) n≥0 ) := e − inf{m≥0:wm =w ′ m } , and the one-sided shift mapθ((w n ) n≥0 ) = (w n+1 ) n≥0 . The resulting topology is generated by the collection of cylinders: Let us introduce the canonical projection Π : Σ →Σ, Π((w n ) n∈Z ) = (w n ) n≥0 . Denote byν the image probability measure (onΣ) of ν by Π. There exists a function ψ :Σ → Z such that ψ • Π = ϕ • θ m0 . let us denote by P : L 2 (ν) → L 2 (ν) the Perron-Frobenius operator such that: Let η ∈]0; 1[. Let us denote by B the set of bounded η-Hölder continuous function g :Σ → C endowed with the usual Hölder norm : We denote by B * the topological dual of B. For all u ∈ R, we consider the operator P u defined on (B, ||.|| B ) by: P u (f ) := P (e iuψ f ).
Note that the hypothesis of non-arithmeticity of ϕ is equivalent to the following one on ψ: for any u ∈ [−π; π]\{0}, the operator P u has no eigenvalue on the unit circle. We will use the method introduced by Nagaev in [8] and [9], adapted by Guivarch and Hardy in [5] and extended by Hennion and Hervé in [7]. It is based on the family of operators (P u ) u and their spectral properties expressed in the two next propositions.
Proposition 3.4. (Uniform Contraction). There exist α ∈ (0; 1) and C > 0 such that, for all u ∈ [−π; π]\[−β; β] and all integer n ≥ 0, for all f ∈ B, we have: This property easily follows from the fact that the spectral radius is smaller than 1 for u = 0. In addition, since P is a quasicompact operator on B and since u → P u is a regular perturbation of P 0 = P , we have : , for all f ∈ B and for all n ≥ 0, we have the decomposition: Lemma 3.6. There exist γ ′ > 0 and C η > 0 such that, ∀q ≥ m 0 and all 2q-cylinderÂ ofΣ, we have: In particular, we haveν(Â) ≤ C η e −γ ′ (2q−m0) . Proof.
Thus, from these computations, we verify that: Now, we treat the second case where n ≤ m 0 . Here, if x ∈θ 2q−m0Â , then y / ∈θ 2q−m0Â , From all this process, setting γ ′ := min(η, − max h) > 0, we get an estimation for the η-Hölder coefficient, ∀n ≥ 0: Hence, for C η := (1 + max (e ηm0 , c(|h| η + |ψ| η ))), we deduce that Next proposition is a two-dimensional version of Proposition 13 in [11]. We give a more precise error term in order to accomodate the one-dimensional case. It may be viewed as a doubly local version of the central limit theorem: first, it is local in the sense that we are looking at the probability that S n ϕ = 0 while the classical central limit theorem is only concerned with the probability that |S n ϕ| ≤ ǫ √ n; second, it is local in the sense that we are looking at this probability conditioned to the fact that we are starting from a set A and landing on a set B on the base. Proposition 3.7. There exist real numbers C 1 > 0 and γ > 0 such that, for all integers n, q, k such that n − 2k > 0 and all m 0 < q ≤ k, all two-sided q-cylinders A of Σ and all measurable subset B ofΣ, we have: Proof. Set Q := A∩{S n ϕ = 0}∩θ −n (θ k (Π −1 (B)). The proof of the proposition will be illustrated in estimating the measure of the set Q.
Since ϕ • θ m0 = ψ • Π and using the semi-conjugacyθ • Π = Π • θ, we have the identity: {S n ϕ • θ m0 = 0} = {S n ψ • Π = 0}. In addition, Im(ψ) ∈ Z, thus we have: Since the measure ν is θ-invariant, then we can verify that: Now we want to estimate the expectation a(u) = Eν(...). Introducing the Perron-Frobenius operator P, and using the fact that it is the dual ofθ, we get: We will treat two cases concerning the values of u. Let us denote for simplicity l := n − (k − m 0 − q). First, using the contraction inequality given in Proposition 3.4 applied to P l u (1), the fact that ||P q u P q (1Â • θ m0 )|| B ≤ e −γ ′ (2q−m0) from Lemma 3.6, and the fact that E P k−m0 u (1 B g) ≤ν(B)||g|| B , we will show that a(u) is negligible for large values of u, so when u / ∈ [−β, β] we get for γ = 2γ ′ : We now use the decomposition in 3.5 to obtain an estimation of the main term coming from small values of u. Indeed, whenever u ∈ [−β, β], we have: Using inequality (1) in Proposition 3.5, one can see that the second term is of order The mappings u → v u and u → φ u are C 1 -regular with v 0 = 1 and ϕ 0 =ν, from which we find that: To obtain an approximation of the first term a 1 (u), we introduce the formula of P in P u : so that, from this approximation, we get: Using Proposition 3.5 and that u → λ u belongs to C 3 ([−β; β] → C), hence applying the intermediate value theorem gives: for the constant c 2 = c 1 /2.
As a consequence, an estimate for a 1 (u) is: ). A final step to reach an estimation of ν(Q) is to integrate the approximated quantity of a 1 (u) obtained above. Using the Gaussian integral, a change of variable v = u √ l gives: In the same way we treat the error term to get: From these computations, it follows that: From this main estimate and (3.1) and (3.4) we conclude that: Proof of the pointwise convergence of the recurrence rate to the dimension. Let us denote by G n (ǫ) the set of points for which n is an ǫ-return : G n (ǫ) := {x ∈ Σ : S n ϕ(x) = 0 and d(θ n (x), x) < ǫ}.
Let us consider the first return time in an ǫ-neighborhood of a starting point x ∈ Σ : Proof of Theorem 3.1. Let us denote by C k the set of two-sided k-cylinders of Σ. For any δ > 0 denote by C δ k ⊂ C k the set of cylinders C ∈ C k such that ν(C) ∈ (e −(d+δ)k , e −(d−δ)k ). For any x ∈ Σ, let C k (x) ∈ C k be the k-cylinder which contains x. Since d is twice the entropy of the ergodic measure ν, by the Shannon-Breiman theorem, the set K δ N = {x ∈ Σ : ∀k ≥ N, C k (x) ∈ C δ k } has a measure ν(K δ N ) > 1 − δ provided N is sufficiently large.
• First, let us prove that, almost surely : Let α > 1 d and 0 < δ < d − 1 α . Let us take ǫ n := n − α 2 and k n := ⌈− log ǫ n ⌉. In view of Proposition 3.7, whenever k n ≥ N , we have : Notice that for ǫ n and k n taken as above, one can verify that the term , from which it follows that Hence by the Borel Cantelli lemma, for a.e. x ∈ K δ N , if n is large enough, we have τ ǫn > n, which in turn implies that : and this proves the the lower bound on the lim inf, since (ǫ n ) n decreases to zero and lim inf n→+∞ ǫn ǫn+1 = 1, and since we have taken an arbitrary α > 1 d . • Next, we will prove the upper bound (d) on the lim sup : let α ∈ (0, 1 d ) and δ > 0 such that 1 − αd − αδ > 0. Take ǫ n := n − α 2 and k n := ⌈− log ǫ n ⌉. We define for all l = 1, ..., n, the sets A l (ǫ) := G l (ǫ) ∩ θ −l {τ ǫ > n − l} which are pairwise disjoint. Setting L n := ⌈n a ⌉ with a > α(d + δ − γ), we then realize that: But due to Proposition 3.7, for any C ∈ C δ kn and l ≥ L n , whenever k n ≥ N , we have : Indeed, the error is negligible, because for a > α(d + δ − γ), Next, we will work to prove that ν K δ N ∩ {τ ǫn > n} is summable. Observe that: But, from (3.5), it follows immediately that . Now let us take n p := p − 4 1−αd−αδ . We have: p≥1 ν(K δ N ∩ {τ ǫn p > n p }) is finite, revealing that, using Borel Cantelli lemma, almost surely x ∈ K δ N ,τ ǫn p (x) ≤ n p , which implies that : This gives the estimate lim sup since (ǫ np ) p decreases to 0 and since lim p→+∞ ǫn p ǫn p+1 = 1.

3.3.
Fluctuations of the rescaled return time. Throughout this subsection, we adapt the general strategy of [12,13]. Recall that C k (x) = {y ∈ Σ : d(x, y) < e −k }. Let R k (y) := min{n ≥ 1 : θ n (y) ∈ C k (y)} denote the first return time of a point y into its k-cylinder C k (y), or equivalently the first repetition time of the first k symbols of y. We recall that C k (x) = {y ∈ Σ : d(x, y) < e −k }. There have been a lot of studies on the quantity R k , among all the results we will use the following.
Proposition 3.8. (Hirata [6]) For ν-almost every point x ∈ Σ, the return time into the cylinders C k (x) are asymptotically exponentially distributed in the sense that for a.e. x, where the convergence is uniform in t.
Proof. Let k ≥ m 0 and n be some integers. We make a partition of a cylinder C k (x) according to the value l ≤ n of the last passage in the time interval 0, ..., n of the orbit of (x, 0) by the map F into C k (x) × {0}. This gives the following equality : . We claim that : According to the decomposition (3.6) and to Proposition 3.7, there exists c 1 > 0 such that we have : Our claim follows from the fact that βν(C k (x)) l=n k l=2 k+1 1 √ l−k ≃ βt and the term k 2 e −γk l=n k l=2 k+1 1 l−2 k ≪ 1.
is tight.
Hence it will be enough to prove that the advertised limit law is the only possible accumulation point of our destination. We hence abbreviate √ τ e −k Lemma 3.11. Suppose that the sequence of conditional distributions of (X kp | C kp (x)) p converges to the law of some random variable X. Then the limit satisfies the integral equation: Proof. To begin with, let us set f (t) := P(X > t) • First, we will prove that The decomposition in (3.6) and Proposition 3.7, implies that there exists c > 0 such that we have: We want to estimate the formula of this inequality when k → ∞. We note that through the proof of Lemma 3.9, it has been proved that lim kp→∞ n kp l=1 we are left to estimate mainly the lower bound on the lim inf of ν(C kp (x))B n kp as p → ∞. Now, by monotonicity, we have Observe that the term: Thus, now by evaluating the following sum: We obtain But, from the hypothesis that P X kp > t 1 − r/N kp→∞ −→ f t 1 − r/N , we get: Combining these estimates and taking the limit when k p → ∞, we establish the desired inequality: • In the same way we treat the converse inequality, using the other half of Proposition 3.7, then there exists c ′ > 0, such that: , then using Proposition 3.8, we have from which we observe that we can forget the first m k term of the following sum, because Furthermore, one verifies that this sum of terms between m k and ⌊n k /N ⌋ is bounded above by 2t ν(C k (x)) √ N . Hence, we get: where N is so large that the last three terms goes to 0 as k → 0. Moreover, if we set B ′ nn k We now proceed to show the bound on the lim sup of ν(C k (x))B ′ n kp as p → 0 It can be easily seen that hence, it follows immediately that Applying lim sup when p → ∞, then lim sup Taking the limit when k p → ∞, and combining all these estimates, we get the second inequality: Lemma 3.12. We know that the conditional distributions of the X kp converge to a random variable X iff the conditional distributions of the X 2 kp converge to X 2 . The later then satisfies Hence, for any s > 0, we find Proof of theorem 3.2. According to Lemma 3.9, the family of distributions of X k is tight. By Lemmas 3.11, 3.12 and 3.13, the law of c β E |N | is the only possible accumulation point of the family of distributions of ν(C k (x)) √ τ e −k | C k (x) k≥0 . Let P be a probability measure absolutely continuous with respect to µ, with density h. Set H(x) := l∈Z h(x, l). Note that by Z-periodicity, the distribution of τ ǫ under P is the same of that under the probability measure with density (x, l) → H(x) with respect to ν ⊗ δ 0 . Assume first that the density H is continuous. Denote by A k := {y : (ν(C k (y) τ e −k (y) > t}, then we have: And so, by the dominated Lebesgue theorem, we get: Now, take in general the density H in L 1 (ν). We use the fact that the set of the continuous functions is dense in L 1 (ν), so that there exists H n continuous such that H n L 1 (ν) −→ H.
We know that there is n such that ∀ ǫ > 0 ||H n − H|| L 1 (ν) < ǫ 2 . Moreover H n is continuous, then there is k such that ∀ ǫ > 0 Hence the conclusion follows.
Proof of Corollary 3.3. Let us set: We have the case that ν is a Gibbs measure with a non degenerate Hölder potential h. There is a constant c h > 0 such that log ν(C k (x)) = k j=−k h • σ j (x). This Birkhoff sum follows a central limit theorem (e.g. [2]), which implies that: log ν(C k (.)) + kd √ k dist −→ N (0, 2σ 2 h ).
Observe that Y k has the following decomposition: Hence, it will be enough to prove that the first term of Y k converges in distribution to 0, which is true due to Theorem 3.2.