A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913682108186624 |
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| author | Sponheimer, Alejandro Rodriguez |
| author_facet | Sponheimer, Alejandro Rodriguez |
| contents | Let $(X,μ,T,d)$ be a metric measure-preserving dynamical system such that $3$-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for $μ$-a.e. $x\in X$ \[ \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} μ(B_k(x))} = 1, \] where $μ(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12928 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms Sponheimer, Alejandro Rodriguez Dynamical Systems 37D20 (Primary) 37A05, 37B20 (Secondary) Let $(X,μ,T,d)$ be a metric measure-preserving dynamical system such that $3$-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for $μ$-a.e. $x\in X$ \[ \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} μ(B_k(x))} = 1, \] where $μ(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions. |
| title | A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms |
| topic | Dynamical Systems 37D20 (Primary) 37A05, 37B20 (Secondary) |
| url | https://arxiv.org/abs/2307.12928 |