Dynamical Borel-Cantelli lemma for recurrence theory

Abstract

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 \[ R(ψ)= \{x\in X : d(T^n x, x) < ψ(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

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