Ergodic actions of countable groups and finite generating partitions

TL;DR

This paper proves that ergodic actions of countable groups with finite Shannon entropy generating partitions necessarily have finite generating partitions, linking entropy properties to the existence of finite partitions.

Contribution

It establishes a connection between finite Shannon entropy generating partitions and finite generating partitions for ergodic group actions, a previously unresolved question.

Findings

Finite Shannon entropy implies the existence of finite generating partitions.

The result applies to ergodic actions of countable groups.

It advances understanding of entropy and partition structure in ergodic theory.

Abstract

We prove that if an ergodic action of a countable group on a probability space admits a generating partition having finite Shannon entropy then it admits a finite generating partition.