Dynamical Borel–Cantelli lemma for recurrence theory

نویسندگان

چکیده

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.

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ژورنال

عنوان ژورنال: Ergodic Theory and Dynamical Systems

سال: 2021

ISSN: ['0143-3857', '1469-4417']

DOI: https://doi.org/10.1017/etds.2021.23