Relative entropy dimensions for amenable group actions

TL;DR

This paper introduces new concepts like relative entropy dimensions and relative dimension sets to analyze the topological complexity of entropy zero extensions under amenable group actions, generalizing previous disjointness results.

Contribution

It defines relative entropy dimensions and related sets for amenable group actions, establishing their properties and relations, and extends disjointness results for entropy zero extensions.

Findings

Defined relative entropy dimensions and generating set dimensions.

Established relations among these dimensions.

Generalized disjointness results for entropy zero extensions.

Abstract

We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Folner sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. Meanwhile, we investigate the relations among them. Secondly, we introduce the notion of a relative dimension set. Moreover, using it, we discuss the disjointness between the relative entropy zero extensions which generalizes the results of Dou, Huang and Park[Trans. Amer. Math. Soc. 363(2) (2011), 659-680].