Dynamical Borel-Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems

TL;DR

This paper explores how dynamical Borel-Cantelli lemmas influence the growth rates of Birkhoff sums for non-integrable observables in various chaotic dynamical systems, providing both general results and specific examples.

Contribution

It introduces new connections between Borel-Cantelli lemmas and growth rates of Birkhoff sums for non-integrable functions in ergodic systems, with applications to different classes of maps.

Findings

Derived growth rate estimates for Birkhoff sums of non-integrable observables.

Established results for non-uniformly expanding and hyperbolic systems.

Provided concrete examples illustrating theoretical implications.

Abstract

We consider implications of dynamical Borel-Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables $\varphi(x) = d(x,p)^{-k}$, $k>0$, on ergodic dynamical systems $(T,X,\mu)$ where $\mu(X) = 1$. Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems.