Topological entropy for nonuniformly continuous maps

Abstract

The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map $f: X \rightarrow X$ from the case when $X$ is a compact metric space to the case when $X$ is allowed to be noncompact. These extensions all require the space $X$ to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When the map $f$ is uniformly continuous, it extends continuously to the completion, and the various notions of entropy reduce to the standard ones (applied to this extension). However, when uniform continuity is not assumed, these new quantities can differ. We consider extensions proposed by Bowen (maximizing over compact subsets and a definition of Hausdorff dimension type) and Friedland (using the compactification of the graph of $f$) as well as a straightforward extension of Bowen and Dinaburg's definition from the compact case, assuming that $X$ is totally bounded, but not necessarily compact. This last extension agrees with Friedland's, and both dominate the one proposed by Bowen (Theorem 6). Examples show how varying the metric outside its uniform class can vary both quantities. The natural extension of Adler--Konheim--McAndrew's original (metric-free) definition of topological entropy beyond compact spaces dominates these other notions, and is unfortunately infinite for a great number of noncompact examples.