Conditional entropy for Amenable group actions

TL;DR

This paper defines and explores the properties of conditional entropy for actions of infinite amenable groups on Lebesgue spaces, including a decomposition theorem, advancing the understanding of entropy in this context.

Contribution

It introduces a new definition of conditional entropy for amenable group actions and establishes fundamental properties and a decomposition theorem for this entropy.

Findings

Defined conditional entropy for amenable group actions

Proved properties of the conditional entropy

Established a decomposition theorem for the entropy

Abstract

Let G be an infinite discrete countable amenable group acting continuously on a Lebesgue space X. In this article, using partition and factor-space, the conditional entropy of the action G is defined. We introduction some properties of conditional entropy for amenable group actions and the corresponding decomposition theorem is obtained.