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Proceedings of the American Mathematical Society

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The nonexistence of expansive homeomorphisms of $ 1$-dimensional compact ANRs


Author: Hisao Kato
Journal: Proc. Amer. Math. Soc. 108 (1990), 267-269
MSC: Primary 54E40; Secondary 54F50, 54H20, 58F15
DOI: https://doi.org/10.1090/S0002-9939-1990-0991698-1
MathSciNet review: 991698
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Abstract: It is well known that if $ X$ is an arc or a circle, then there is no expansive homeomorphism on $ X$ (see [2] and [3]). In this note, we show that if $ X$ is a Peano continuum which has a neighborhood $ M$ such that $ {\text{cl}}\left( M \right)$ is a $ 1$-dimensional AR, then there is no expansive homeomorphism on $ X$ . In particular, no $ 1$-dimensional compact ANR admits an expansive homeomorphism.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0991698-1
Keywords: Expansive homeomorphism, dendrite (=$ 1$-dimensioinal compact AR), $ 1$-dimensional compact ANR
Article copyright: © Copyright 1990 American Mathematical Society