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

我们研究了具有兼容度量d的保测动态系统 $(X, \mu, T)$ 中递归集的动态 Borel-Cantelli 引理。我们证明在一些正则性条件下，以下集合 $$ \begin{align*}R(\psi)= \{x\in X : d(T^nx, x) < \ psi(n)\ \text{对于无穷多}\ n\in\mathbb{N} \} \end{align*}$$ 根据某个级数的收敛或发散遵循零满定律，其中 $ \psi :\mathbb {N}\to \mathbb {R}^+$ 。我们的主要定理的应用包括高斯映射、 $\beta $ -变换和齐次自相似集。