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Transactions of the American Mathematical Society

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



Expansive homeomorphisms and topological dimension

Author: Ricardo Mañé
Journal: Trans. Amer. Math. Soc. 252 (1979), 313-319
MSC: Primary 58F15; Secondary 54H20
MathSciNet review: 534124
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Abstract: Let K be a compact metric space. A homeomorphism $ f:\,K\mid$ is expansive if there exists $ \varepsilon \, > \,0$ such that if $ x, y\, \in \,K$ satisfy $ d\left( {{f^n}\left( x \right),\,{f^n}\left( y \right)} \right)\, < \,\varepsilon $ for all $ n\, \in \,{\textbf{Z}}$ (where $ d\left( { \cdot ,\, \cdot } \right)$ denotes the metric on K) then $ x\, = \,y$. We prove that a compact metric space that admits an expansive homeomorphism is finite dimensional and that every minimal set of an expansive homeomorphism is 0-dimensional.

References [Enhancements On Off] (What's this?)

  • [1] R. Bowen, Markov partitions and minimal sets for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 903-918. MR 0277002 (43:2739)
  • [2] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1948. MR 0006493 (3:312b)

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Article copyright: © Copyright 1979 American Mathematical Society

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