Pointwise convergence of ergodic averages in Orlicz spaces

Abstract

We construct a sequence ${a_n}$ such that for any aperiodic measure-preserving system $(X, \Sigma, m, T)$ the ergodic averages \begin{equation*} A_Nf(x) = \frac{1}{N} \sum_{n=1}^N f\bigl(T^{a_n}x\bigr) \end{equation*} converge a.e. for all $f$ in $L \log\log(L)$ but fail to have a finite limit for an $f \in L^1$. In fact, 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 $f \in L^1$. Our method, introduced by A. Bellow and extended by K. Reinhold and M. Wierdl, is perturbation.