Kac's Lemma and countable generators for actions of countable groups

TL;DR

This paper generalizes Kac's lemma to actions of countable groups and introduces the concept of allocation, leading to new results on generating partitions in ergodic theory.

Contribution

It introduces the notion of allocation for countable group actions and generalizes Kac's lemma to these actions and equivalence relations.

Findings

Generalized Kac's lemma for countable group actions

Proved existence of countable generating partitions for ergodic actions

Provided a new proof technique using allocations

Abstract

Kac's lemma determines the expected return time to a set of positive measure under iterations of an ergodic probability preserving transformations. We introduce the notion of an \emph{allocation} for a probability preserving action of a countable group. Using this notion, we formulate and prove generalization of Kac's lemma for an action of a general countable group, and another generalization that applies to probability preserving equivalence relations. As an application, we provide a short proof for the existence of countable generating partitions for any ergodic action of a countable group.