# Convergence in distribution for Z-extension of Axiom A flow

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## Proof of the pointwise convergence of the recurrence rate to the dimension.

We suppose that the initial point M0 is in (0, 1)d. Let ǫ > 0 so small that B(M0, ǫ) is contained in (0, 1)d, then Leb(B(x, ǫ)) = cǫd. Definition. Setting R0 = 0, let us first define R1 = min{m ≥ 1 : Sm = 0}. We then define by induction, for any p ≥ 0 the pth return time Rp of (Mn)n≥0 in (0, 1)d, by Rp+1 := inf m > Rp : Sm = 0 Definition. We define the return time Tǫ of the sequence of random variables (YRn)n≥0 in the ǫ−neighborhood B(Y0, ǫ), by: Tǫ := min{l ≥ 1 : YRl ∈ B(Y0, ǫ)}. Thus, we have the following relation τǫ = RTǫ . (1.2.1) We will study the asymptotic behavior of the random variables Rn and Tǫ and use the relation (1.2.1) to prove Theorem 1.1.1.

### Behavior of the random variable Rn.

Lemma 1.2.1. P n≥0 P(S2n = 0)s2n = 1 √1−s2 and P(S2n = 0) ∼ 1 √πn. Proof. Recall that (Xi)i is a sequence of independent random variables such that P(Xi = 1) = P(Xi = −1) = 1/2. Note that (Xi+1) 2 is equal to 1 if Xi = 1, and is equal to 0 if Xi = −1. Hence (Xi+1) 2 is a random variable of Bernoulli distribution of parameter 1 2 .

#### Spectral Analysis of the Perron-Frobenius operator.

We define the one-sided shift ˆ := {w := (wn)n∈N ∈ AN : ∀n ∈ N,M(wn,wn+1) = 1}.
As we did for , we endow ˆ with the metric ˆ d defined by ˆ d((wn)n≥0, (w ′ n)n≥0) := e−ˆr(w,w ′ )  with ˆr((wn)n≥0, (w′ n)n≥0) = inf{m ≥ 0 : wm 6= w′ m}. Moreover, the shift ˆθ is the restriction on ˆ of the one-sided shift defined by ˆθ((wn)n≥0) = (wn+1)n≥0.
Remark 2.1.1. For any H¨older function ˜ f defined on ˆ, ˜ f : ˆ R → R, such that ˜ fdˆν = 0, we denote its ergodic sum by ˆ Sn ˜ f = Xn−1 l=0 ˜ f ◦ ˆθl. Let us define the canonical projection : → ˆ defined by ((wn)n∈Z) = (wn)n≥0. Let ˆν be the image probability measure (on ˆ) of ν by . Let ω ∈ . Since ϕ is constant on the m0-cylinder, ϕ ◦ θm0ω depends only on ω, then there exists a function ψ : ˆ → Z such that ψ ◦ = ϕ ◦ θm0 .

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Proof of the Local Limit Theorem

Next proposition is very essential in this work. 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 Snϕ = 0 while the classical central limit theorem is only concerned with the probability that |Snϕ| ≤ 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. Proposition 2.2.1. There exist real numbers C1 > 0 and γ > 0 such that, for all integers n, q, q′ , k such that n − 2k ≥ m0 and m0 < q ≤ k, for all (q, q′)-cylinders A of and all measurable subset B of ˆ, we have.

Introduction
Introduction
1 Recurrence in a Probabilistic Toy Model
1.1 Description of the model and statements of the results.
1.2 Proof of the pointwise convergence of the recurrence rate to the dimension.
1.2.1 Behavior of the random variable Rn.
1.2.2 Behavior of the random variable Tǫ
1.3 Proof of the convergence in distribution of the rescaled return time.
2 Local Limit Theorem with speed for subshift of finite type
2.1 Spectral Analysis of the Perron-Frobenius operator.
2.2 Proof of the Local Limit Theorem
3 Recurrence for Z-extension of subshift of finite type.
3.1 Description of the Z-extension of a mixing subshift and statement of the results
3.2 Proof of the pointwise convergence of the recurrence rate to the dimension
3.3 Fluctuations of the rescaled return time.
4 Properties of Axiom A flows
4.1 Definition of Axiom A Flows
4.2 Markov Sections
4.3 Representation by a special flow over a subshift
4.4 Suspension Flow
4.5 Equilibrium Measures
4.5.1 Equilibrium measures for the flows
4.5.2 Equilibrium measures for symbolic suspension flows
4.6 Balls and Coding
5 Pointwise convergence of the recurrence rate to the dimension
5.1 Description of the Z-extension
5.2 Proof of the almost sure convergence Theorem
6 Convergence in distribution for Z-extension of Axiom A flow
6.1 Construction of the partition
6.2 Proof of the convergence in distribution .

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