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Lowering topological entropy over subsets for amenable group actions

Xiaochen Wang

TL;DR

The paper develops a comprehensive framework for lowering topological entropy along subsets in countable amenable group actions, introducing notions such as lowerable, D-lowerable, P-lowerable, and their hereditary counterparts. It proves that systems with finite topological entropy are lowerable in multiple senses, and establishes a fundamental equivalence between asymptotic $h$-expansiveness and hereditary uniform lowerability, via a Bowen-type entropy theorem for amenable groups. The results are extended through mass-distribution principles linking subset entropy to ergodic components, and the paper demonstrates that these lowerability properties are preserved under principal extensions. Additional contributions include handling expansive systems, and constructing principal extensions to transfer entropy-lowering properties, with implications for understanding entropy structure in amenable group actions and their factors.

Abstract

In this paper, we introduce the notions of lowerable, D-lowerable, P-lowerable, hereditarily lowerable, and hereditarily uniformly lowerable for countably infinite amenable group actions. We show that a system with finite entropy is lowerable, D-lowerable, and P-lowerable, and that asymptotic h-expansiveness is equivalent to hereditary uniform lowerability. Moreover, we prove a Bowen's type theorem for amenable group actions.

Lowering topological entropy over subsets for amenable group actions

TL;DR

The paper develops a comprehensive framework for lowering topological entropy along subsets in countable amenable group actions, introducing notions such as lowerable, D-lowerable, P-lowerable, and their hereditary counterparts. It proves that systems with finite topological entropy are lowerable in multiple senses, and establishes a fundamental equivalence between asymptotic -expansiveness and hereditary uniform lowerability, via a Bowen-type entropy theorem for amenable groups. The results are extended through mass-distribution principles linking subset entropy to ergodic components, and the paper demonstrates that these lowerability properties are preserved under principal extensions. Additional contributions include handling expansive systems, and constructing principal extensions to transfer entropy-lowering properties, with implications for understanding entropy structure in amenable group actions and their factors.

Abstract

In this paper, we introduce the notions of lowerable, D-lowerable, P-lowerable, hereditarily lowerable, and hereditarily uniformly lowerable for countably infinite amenable group actions. We show that a system with finite entropy is lowerable, D-lowerable, and P-lowerable, and that asymptotic h-expansiveness is equivalent to hereditary uniform lowerability. Moreover, we prove a Bowen's type theorem for amenable group actions.
Paper Structure (14 sections, 34 theorems, 268 equations)

This paper contains 14 sections, 34 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological Entropy for G-Systems and Their Subsets
  4. Sep, spn-entropy of $(X,G)$ and subsets
  5. Bowen topological entropy of $(X,G)$ and subsets
  6. Packing topological entropy of $(X,G)$ and subsets
  7. Properties of Lowerability for $G$-systems with Finite Entropy
  8. Distribution Principles In $G$-system
  9. Lowerability, $D$-Lowerability, and $P$-Lowerability of $G$-Systems with Finite Entropy
  10. Properties of Lowerability for an Expansive $G$-system
  11. The Equivalence Between Asymptotically $h$-Expansiveness and Hereditarily Uniformly Lowerability for Dynamical Systems
  12. Asymptotic $h$-Expansivity of Hereditarily Uniformly Lowerable $G$-Systems
  13. Hereditary Uniform Lowerability of Asymptotically $h$-Expansive $G$-Systems
  14. Principal Extensions: Preserving Lowerable, Hereditarily Lowerable, and Hereditarily Uniformly Lowerable Properties

Key Result

Proposition 2.1

Let $(X,G)$ be a $G$-system, $\{F_n\}_{n \in \mathbb{N}}$ a Følner sequence for $G$, and $\mathcal{U} \in \mathcal{C}_X^o$. Then:

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 55 more