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 three-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, that is, for $\mu $ -almost every $x\in X$ , $$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$ 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.