Divergence of averages obtained by sampling a flow

Abstract:

In this paper we consider ergodic averages obtained by sampling at discrete times along a measure preserving ergodic flow. We show, in particular, that if ${U_t}$ is an aperiodic flow, then averages obtained by sampling at times $n + {t_n}$ satisfy the strong sweeping out property for any sequence ${t_n} \to 0$. We also show that there is a flow (which is periodic) and a sequence ${t_n} \to 0$ such that the Cesaro averages of samples at time $n + {t_n}$ do converge a.e. In fact, we show that every uniformly distributed sequence admits a perturbation that makes it a good Lebesgue sequence.