Slow entropy for noncompact sets and variational principle

TL;DR

This paper introduces a new concept of slow entropy for noncompact sets in Z^d actions, establishing a variational principle and exploring its relation to Bowen entropy through examples and theoretical analysis.

Contribution

It defines topological and measure-theoretic slow entropy for Z^d actions and proves a variational principle linking these notions with Bowen topological entropy.

Findings

Established a variational principle for slow entropy.

Provided examples illustrating the computation of slow entropy.

Analyzed the relationship between slow entropy and Bowen entropy.

Abstract

This paper defines and discusses the dimension notion of topological slow entropy of any subset for Z^d actions. Also, the notion of measure-theoretic slow entropy for Z^d actions is presented, which is modified from Brin and Katok [2]. Relations between Bowen topological entropy [3,17] and topological slow entropy are studied in this paper, and several examples of the topological slow entropy in a symbolic system are given. Specifically, a variational principle is proved.