Entropy on quasi-uniform spaces

Abstract

Quasi-uniform entropy \(h_{QU}(\psi)\) is defined for a uniformly continuous self-map \(\psi\) on a \(T_0\) quasi-uniform space \((X,\mathcal{U})\). Basic properties are proved about this entropy, and it is shown that the quasi-uniform entropy \(h_{QU}(\psi ,\mathcal{U})\) is less than or equal to the uniform entropy \(h_U(\psi, \mathcal{U}^s)\) of \(\psi\) considered as a uniformly continuous self-map of the uniform space \((X,\mathcal{U}^s)\), where \(\mathcal{U}^s\) is the uniformity associated with the quasi-uniformity \(\mathcal{U}\). Finally, we prove that the completion theorem for quasi-uniform entropy holds in the class of all join-compact \(T_0\) quasi-uniform spaces, that is for join-compact \(T_0\) quasi-uniform spaces the entropy of a uniformly continuous self-map coincides with the entropy of its extension to the bicompletion.