New and old facts about entropy in uniform spaces and topological groups

Abstract

In 1965 Adler, Konheim and McAndrew defined the topological entropy of a continuous self-map of a compact space. In 1971 Bowen extended this notion to uniformly continuous self-maps of (not necessarily compact) metric spaces and this approach was pushed further to uniform spaces and topological groups by many authors, giving rise to various versions of the topological entropy function. In 1981 Peters proposed a completely different (algebraic) entropy function for continuous automorphisms of non-compact LCA groups. The aim of this paper is to discuss some of these notions and their properties, trying to describe the relations among the various entropies and to correct some errors appearing in the literature.