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

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.