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Dynamical Borel–Cantelli lemma for recurrence theory
Ergodic Theory and Dynamical Systems ( IF 0.8 ) Pub Date : 2021-04-12 , DOI: 10.1017/etds.2021.23
MUMTAZ HUSSAIN 1 , BING LI 2 , DAVID SIMMONS 3 , BAOWEI WANG 4
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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.



中文翻译:

递归理论的动态 Borel-Cantelli 引理

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。

更新日期:2021-04-12
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