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arXiv · 2307.12928

A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms

Alejandro Rodriguez Sponheimer
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Abstract

Let $(X,\mu,T,d)$ 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 $\mu$-a.e. $x\in X$ \[ \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \] where $\mu(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

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Alejandro Rodriguez Sponheimer. 2023-07-24. A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms. https://doi.org/10.1017/etds.2024.64

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