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. $x\in X$ $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sum _{k=1}^n{\mathbf{1}}_{B_k(x)}(T^{k}x)}{\sum _{k=1}^n\mu (B_k(x))}=1,$$ 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.