Dynamical Borel–Cantelli lemma for recurrence theory

We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system (X, μ, T ) with a compatible metric d. We prove that under some regularity conditions, the μ-measure of the following set R(ψ) = {x ∈ X : d(T nx, x) < ψ(n) for infinitely many n ∈ N} obeys a zero–full law according to the convergence or divergence of a certain series, where ψ : N → R+. The applications of our main theorem include the Gauss map, β-transformation and homogeneous self-similar sets.