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 align*R(ψ)= ∈ X : d(Tⁿ x, x) < ψ(n)\ for infinitely many\ n \ align* 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.