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

TL;DR

This paper establishes a strong recurrence result for Axiom A diffeomorphisms with exponential decay of correlations, extending Borel--Cantelli lemmas to dynamical systems with specific measure decay conditions.

Contribution

It proves a strong dynamical Borel--Cantelli lemma for Axiom A diffeomorphisms under exponential decay of correlations, a novel extension in recurrence theory.

Findings

Validates the strong Borel--Cantelli result for Axiom A systems

Demonstrates the result for equilibrium states under certain conditions

Shows the result holds for sequences converging slowly to zero

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.