Quasi-uniform entropy vs topological entropy

Abstract

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy $h_{QU}\left(\psi \right)$ 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 $h,h_f$ and the quasi-uniform entropy function $h_{QU}$ on a quasi-uniform space X, where h and $h_f$ 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 $T_0$-quasi-uniform space $(X,U)$ we have $h\left(\psi \right)\le h_{QU}\left(\psi \right)$ when X is compact and $h_{QU}\left(\psi \right)\le h_f\left(\psi \right)$ with equality if X is a compact $T_2$ space.