The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circlelike continua
Author:
Hisao Kato
Journal:
Trans. Amer. Math. Soc. 349 (1997), 36453655
MSC (1991):
Primary 54H20, 54F50; Secondary 54E50, 54B20
MathSciNet review:
1401776
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Abstract: A homeomorphism of a compactum with metric is expansive if there is such that if and , then there is an integer such that . It is wellknown that adic solenoids () admit expansive homeomorphisms, each is an indecomposable continuum, and cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each , does there exist a plane continuum so that admits an expansive homeomorphism and separates the plane into components? For the case , the typical plane continua are circlelike continua, and every decomposable circlelike continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circlelike continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudoarcs constructed by W. Lewis and all decomposable circlelike continua, admits no expansive homeomorphisms. In particular, any decomposable circlelike continuum admits no expansive homeomorphism. Also, we show that if is an expansive homeomorphism of a circlelike continuum , then is itself weakly chaotic in the sense of Devaney.
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Additional Information
Hisao Kato
Affiliation:
Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan
Email:
hisakato@sakura.cc.tsukuba.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994797018503
PII:
S 00029947(97)018503
Keywords:
Expansive homeomorphism,
decomposable,
chainable,
circlelike,
the pseudoarc,
pattern,
hyperspace
Received by editor(s):
October 9, 1995
Received by editor(s) in revised form:
February 6, 1996
Article copyright:
© Copyright 1997
American Mathematical Society
