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Dynamical Borel–Cantelli lemma for recurrence theory

Part of: Ergodic theory Measure-theoretic ergodic theory

Published online by Cambridge University Press:  12 April 2021

MUMTAZ HUSSAIN ,
BING LI ,
DAVID SIMMONS  and
BAOWEI WANG
MUMTAZ HUSSAIN
Affiliation:
La Trobe University, P.O. Box 199, Bendigo, VIC 3552, Australia (e-mail: M.Hussain@latrobe.edu.au)
BING LI*
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou 510640, China
DAVID SIMMONS
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (e-mail: David.Simmons@york.ac.uk)
BAOWEI WANG
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China (e-mail: bwei_wang@hust.edu.cn)
*
e-mail: scbingli@scut.edu.cn
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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

$$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$
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.

Keywords

recurrencedynamical Borel-Cantelli lemmazero-full lawexponentially mixing

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 28D05: Measure-preserving transformations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 1994 - 2008
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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