The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua |
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| Authors: | Hisao Kato |
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| Affiliation: | Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan |
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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 well-known 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 circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if  is an expansive homeomorphism of a circle-like continuum  , then  is itself weakly chaotic in the sense of Devaney. |
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| Keywords: | Expansive homeomorphism decomposable chainable circle-like the pseudo-arc pattern hyperspace |
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