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

TL;DR

The paper generalizes Kac's lemma from single ergodic transformations to probability-preserving actions of countable groups and to probability-preserving equivalence relations by introducing $A$-allocations. It proves the fundamental identity $\int_A f_\kappa(x)\,d\mu(x)=\int_X f(x)\,d\mu(x)$ with the key corollary $\int_A |B_\kappa(x)|\,d\mu(x)=1$, and extends these ideas to $\mathbb{Z}^d$ with almost-convex shapes, as well as to equivalence relations via orbit representations. The paper culminates in a concise demonstration that every ergodic action of a countable group admits a countable generator, with a broader result also applicable to actions with infinite orbits. These results unify return-time phenomena across group actions and provide structural tools (generators) for ergodic theory and orbit-equivalence contexts.

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.

Theorems & Definitions (25)

• Theorem 1.1: Kac's lemma
• Definition 1.2
• Proposition 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Proposition 1.7
• Theorem 1.8: Kac's lemma for probability preserving equivalence relations
• Lemma 2.1
• proof