Quasi-uniform entropy vs topological entropy (Q6611784)

In [Topology Appl. 332, Article ID 108512, 17 p. (2023; Zbl 1517.54007); Acta Math. Hung. 171, No. 2, 241--266 (2023; Zbl 07794381)] \textit{P. Haihambo} and \textit{O. Olela-Otafudu} introduced the notion of quasi-uniform entropy \(h_{QU}(\psi)\) for a uniformly continuous self-map \(\psi\) of a quasi metric (uniform) space \(X\), extending the classical notion of topological entropy \(h_U\) for uniformly continuous self-maps on metric spaces given by \textit{R. Bowen} [Trans Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)]. In this paper, they deal with the relationship between this notion and the topological entropy using compact sets, \(h\), and the topological entropy using finite open covers, \(h_f\). Let \((X,\mathcal{U})\) be a quasi-uniform space and \(\psi\) be a uniformly continuous self-map on \((X,\mathcal{U})\), then it is proved (Theorem 1) that if \((X,\mathcal{U})\) is compact then \(h(\psi,\tau(\mathcal{U}))\leq h_{QU}(\psi, \mathcal{U})\), whereas in Theorem 2 it is proved that if \((X,\mathcal{U})\) is a \(T_2\)-space then \(h_{QU}(\psi,\mathcal{U})\leq h_f(\psi,\tau(\mathcal{U}))\), where \(\tau(U)\) is the topology induced by the uniformity \(\mathcal{U}\). Moreover, an example in which this last inequality is strict is given. Finally, it is proved that if \((X,\mathcal{U})\) is a compact-\(T_2\)-quasi-uniform space and \(\psi\) as before then the three notions are equal, that is, \(h(\psi,\tau(\mathcal{U}))=h_{QU}(\psi,\mathcal{U})=h_f(\psi,\tau(\mathcal{U}))\).