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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Entropy-expansive maps

Author: Rufus Bowen
Journal: Trans. Amer. Math. Soc. 164 (1972), 323-331
MSC: Primary 28.70; Secondary 54.00
MathSciNet review: 0285689
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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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Keywords: Entropy, h-expansive
Article copyright: © Copyright 1972 American Mathematical Society