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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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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Keywords: Expansive homeomorphism, dendrite (=<IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$1$">-dimensioinal compact AR), <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$1$">-dimensional compact ANR
Article copyright: © Copyright 1990 American Mathematical Society