Upper capacity entropy and packing entropy of saturated sets for amenable group actions

TL;DR

This paper investigates the upper capacity and packing entropy of saturated sets in topological systems with amenable group actions, establishing that upper capacity entropy always reaches full entropy and providing a variational principle for packing entropy.

Contribution

It introduces new results on entropy properties of saturated sets under amenable group actions, including a variational principle for packing entropy.

Findings

Upper capacity entropy always attains full entropy.

A variational principle for packing entropy of saturated sets is established.

The study extends entropy analysis to systems with weaker specification properties.

Abstract

Let $(X,G)$ be a $G$-action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets.