Recurrence for measurable semigroup actions

TL;DR

This paper investigates the recurrence properties of finitely generated free semigroups of measurable maps, exploring conditions under which recurrence occurs and how it relates to invariant measures and Markov processes.

Contribution

It introduces abstract conservativity conditions for recurrence in semigroup actions and analyzes the inheritance of recurrence between generators and the entire semigroup.

Findings

Almost all points can be recurrent despite wandering sets of positive measure.

Inheritance of recurrence between generators and the whole semigroup can be unexpected.

A reduction to Markov process analysis provides new insights into recurrence properties.

Abstract

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.