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Conditional entropy for Amenable group actions

Yuan Lian, Bin Zhu

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.

Conditional entropy for Amenable group actions

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.
Paper Structure (4 sections, 8 theorems, 31 equations)

This paper contains 4 sections, 8 theorems, 31 equations.

Table of Contents

  1. Factor space
  2. Group actions
  3. Mean conditional entropy
  4. Conditional entropy for amenable group actions

Key Result

Proposition 3.1

Let $G\curvearrowright^{T} (X, \mu)$ be a p.m.p. action on a Lebesgue space $(X,\mathcal{B},\mu)$, $\gamma$ be a measurable paitition, then

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • Definition 4.4
  • ...and 8 more