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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
DOI: https://doi.org/10.1090/S0002-9947-1979-0534124-9
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.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0534124-9
Article copyright: © Copyright 1979 American Mathematical Society