
Abstract
The literature contains several extensions of the standard definitions of
topological entropy for a continuous selfmap $f: X \rightarrow X$
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
AdlerKonheimMcAndrew's original (metricfree) definition of topological
entropy beyond compact spaces dominates these other notions, and is
unfortunately infinite for a great number of noncompact examples.
Mathematics Subject Classification: Primary: 37B40.
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