Strongly recurrent transformation groups

Abstract

We define the notions of strong and strict recurrency for actions of countable ordered groups on σ-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.