Entropy-Expansive Maps

Abstract

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