Pointwise Convergence of Ergodic Averages in Orlicz Spaces

TL;DR

This paper demonstrates that in certain Orlicz spaces, ergodic averages can converge along specific sequences for functions within the space but diverge for all functions in L^1, extending previous convergence results.

Contribution

It extends prior work by showing pointwise convergence of ergodic averages in Orlicz spaces contained in L^1 using a perturbation method.

Findings

Existence of sequences where ergodic averages converge in Orlicz spaces but diverge in L^1.

Extension of Bellow and Reinhold's methods to Orlicz spaces.

Application of perturbation techniques to establish convergence/divergence results.

Abstract

We show that for each Orlicz space properly contained in L^1 there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f in L^1. This extends the work of K. Reinhold, who, building on the work of A. Bellow,constructed a sequence for which the averages converge a.e. for every f in L^p, p>q, but diverge for some f in L^q. Our method, introduced by Bellow and extended by Reinhold and M. Wierdl, is perturbation.