Entropy-expansive maps

Bowen, Rufus

doi:10.1090/S0002-9947-1972-0285689-X

Entropy-expansive maps
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by Rufus Bowen

Trans. Amer. Math. Soc. 164 (1972), 323-331

DOI: https://doi.org/10.1090/S0002-9947-1972-0285689-X

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Abstract:

Let $f:X \to X$ be a uniformly continuous map of a metric space. f is called h-expansive if there is an $\varepsilon > 0$ so that the set ${\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon$ for all $n \geqq 0$} has zero topological entropy for each $x \in X$. For X compact, the topological entropy of such an f is equal to its estimate using $\varepsilon :h(f) = h(f,\varepsilon )$. If X is compact finite dimensional and $\mu$ an invariant Borel measure, then ${h_\mu }(f) = {h_\mu }(f,A)$ for any finite measurable partition A of X into sets of diameter at most $\varepsilon$. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.

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