Quasi-uniform entropy vs topological entropy

Abstract

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy hQU(ψ) for a uniformly continuous self-map ψ of a quasi-metric or a quasi-uniform space X. In this paper, we discuss the connection between the topological entropy functions h,hf and the quasi-uniform entropy function hQU on a quasi-uniform space X, where h and hf are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map ψ of a T0-quasi-uniform space (X,U) we have h(ψ)≤hQU(ψ) when X is compact and hQU(ψ)≤hf(ψ) with equality if X is a compact T2 space.