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Let $(X,\mu,T,d)$ be a metric measure-preserving dynamical system such that$3$-fold correlations decay exponentially for H\"older continuous observables.Given a sequence $(M_k)$ that converges to $0$ slowly enough we obtain a strongdynamical 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 certainequilibrium states.

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