On almost-sure versions of classical limit theorems for dynamical systems

Abstract

The purpose of this article is to support the idea that whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average. We follow three different approaches: martingale methods, spectral methods and induction arguments. Our results apply, among others, to Axiom A maps or flows, to systems inducing a Gibbs-Markov map, and to the stadium billiard.