On strong forms of the Borel--Cantelli lemma and intermittent interval maps

TL;DR

This paper develops new variants of the Borel--Cantelli lemma and applies them to analyze statistical properties of intermittent maps with invariant measures, showing certain sequences of intervals satisfy strong Borel--Cantelli properties.

Contribution

It introduces novel variants of the quantitative Borel--Cantelli lemma and demonstrates their application to intermittent maps with absolutely continuous invariant measures.

Findings

Sequences of intervals with endpoints away from zero are strong Borel--Cantelli sequences

New variants of the Borel--Cantelli lemma are derived for dynamical systems

Applications to statistical properties of intermittent maps

Abstract

We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel--Cantelli sequence with respect to such map and invariant measure.