Dynamical Borel–Cantelli lemma for recurrence theory

Abstract We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving system $(X, \mu , T)$ with compatible metric d . prove that under some regularity conditions, $\mu $ -measure of following set $$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$ obeys zero–full law according to convergence or divergence certain series, where $\psi :\mathbb {N}\to \mathbb {R}^+$ The applications our main theorem include Gauss map, $\beta -transformation and homogeneous self-similar sets.