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

Let (X,μ ,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 μ -almost every x∈ X , align* limn → ∞∑ₖ₌₁ⁿ 1Bₖ(x)(Tᵏx) ∑ₖ₌₁ⁿ μ(Bₖ(x)) = 1, align* where μ (Bₖ(x)) = Mₖ . In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.