A generalized multiple Borel-Cantelli lemma in dynamics and its applications

Sixu Liu

doi:10.3934/dcds.2025079

Article Contents

2025, Volume 45, Issue 12: 4927-4966. Doi: 10.3934/dcds.2025079

This issue Previous Article Liouville theorems and gradient estimates for positive solutions to a class of quasilinear elliptic equations Next Article Indecomposable continua for unbounded itineraries of exponential maps

A generalized multiple Borel-Cantelli lemma in dynamics and its applications

Sixu Liu

Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China

Received: August 2024

Revised: April 2025

Early access: June 3, 2025

Published online: June 3, 2025

The author is supported by Natural Science Foundation of Beijing No. 1244041.

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Abstract

The multiple Borel-Cantelli lemma is a criterion that characterizes the occurrence of multiple rare events on the same time scale. In this work, we extend the version of the multiple Borel-Cantelli lemma in dynamics, as established by Dolgopyat, Fayad and Liu [J. Mod. Dyn. 18 (2022) 209-289], thus broadening its applicability to several non-smooth systems preserving absolutely continuous measures. Using this generalization, we establish multiple logarithm laws for hitting time and recurrence for dispersing billiard maps and piecewise expanding maps under certain regular conditions, including tent maps, Lorenz-like maps and the Gauss map.

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Mathematics Subject Classification: Primary: 37A50; Secondary: 37C83, 60F20.

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