Metric entropy of homeomorphism on non-compact metric space (Q656170)

For a locally compact metric space \( (X,d)\) and a uniformly continuous homeomorphism \(T:X\to X\), \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)] introduced the notion of topological entropy \(h_d (T)\) extending the case of \(X\) being compact. This has been important in particular in allowing entropy to be lifted to non-compact covering spaces in several settings. Such a map induces in a natural way a homeomorphism \(T^{*}:X^{*}\to X^{*}\) of the one-point compactification \(X^{*}\) of \(X\), by defining \(T^{*}\) to be \(T\) on the embedded copy of \(X\) and to fix the extra point. It is shown here that \(h (T^{*})\leq h_d (T)\), where \(h (T^{*})\) denotes the usual topological entropy in the compact setting defined using open covers by \textit{R. L. Adler}, \textit{A. G. Konheim} and \textit{M. H. McAndrew} [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)]. (It is clear that the inequality may be strict, since, for example, if \(T\) is any homeomorphism of the real line then \(T^{*}\) is a homeomorphism of the circle which automatically has zero entropy.)