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.
Citation
Andrew Parrish. "Pointwise convergence of ergodic averages in Orlicz spaces." Illinois J. Math. 55 (1) 89 - 106, Spring 2011. https://doi.org/10.1215/ijm/1355927029
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