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 {U_t} is an aperiodic flow, then averages obtained by sampling at times n + t n n + {t_n} satisfy the strong sweeping out property for any sequence t n → 0 {t_n} \to 0 . We also show that there is a flow (which is periodic) and a sequence t n → 0 {t_n} \to 0 such that the Cesaro averages of samples at time n + t n 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.