A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms

Abstract Let (X, , T, d) be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence (Mₖ) that converges to 0 slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for -almost every x X, align* ₍ ₊=₁^₍ 1₁䂵 (ₗ) (T^{kx) } ₊=₁^{n (Bₖ (x) ) } = 1, align* where (Bₖ (x) ) = Mₖ. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.