Lowering topological entropy over subsets for amenable group actions

TL;DR

This paper introduces new concepts related to entropy in amenable group actions, establishes their relationships, and proves a Bowen's type theorem, advancing the understanding of entropy properties in dynamical systems.

Contribution

It defines several new entropy-related notions for amenable group actions and proves their equivalences and properties, including a Bowen's type theorem.

Findings

Finite entropy systems are lowerable, D-lowerable, and P-lowerable.

Asymptotic h-expansiveness is equivalent to hereditary uniform lowerability.

A Bowen's type theorem is established for amenable group actions.

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.