Generalized multiple Borel-Cantelli Lemma in dynamics and its applications

TL;DR

This paper generalizes the multiple Borel-Cantelli Lemma for dynamical systems, enabling new results on recurrence and hitting times in non-smooth systems like billiards and expanding maps.

Contribution

It extends the lemma to non-smooth systems with absolutely continuous measures, broadening its applicability in dynamical systems theory.

Findings

Derived multiple Logarithm Law for dispersing billiard maps

Established recurrence results for piecewise expanding maps

Extended Borel-Cantelli criteria to non-smooth dynamical systems

Abstract

Multiple Borel-Cantelli Lemma is a criterion that characterizes the occurrence of multiple rare events on the same time scale. We generalize the multiple Borel-Cantelli Lemma in dynamics established by Dolgopyat, Fayad and Liu [J. Mod. Dyn. 18 (2022) 209--289], broadening its applications to encompass several non-smooth systems with absolute continuous measures. Utilizing this generalization, we derive multiple Logarithm Law for hitting time and recurrence of dispersing billiard maps and piecewise expanding maps under some regular conditions, including tent map, Lorentz-like map and Gauss map.