Topological and Metric Recurrence for General Markov Chains

Abstract

Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non recurrent points is of zero reference measure. Necessary and sufficient conditions for a reference measure $m$ (which needs not to be dynamically invariant) to satisfy this property are obtained. These results are new even in the purely deterministic setting.