The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua

Kato, Hisao

doi:10.1090/S0002-9947-97-01850-3

The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua
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Trans. Amer. Math. Soc. 349 (1997), 3645-3655 Request permission

Abstract:

A homeomorphism $f:X \to X$ of a compactum $X$ with metric $d$ is expansive if there is $c > 0$ such that if $x, y \in X$ and $x \not = y$, then there is an integer $n \in \mathbf {Z}$ such that $d(f^{n}(x),f^{n}(y)) > c$. It is well-known that $p$-adic solenoids $S_p$ ($p\geq 2$) admit expansive homeomorphisms, each $S_p$ is an indecomposable continuum, and $S_p$ cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each $1 \leq n \leq 3$, does there exist a plane continuum $X$ so that $X$ admits an expansive homeomorphism and $X$ separates the plane into $n$ components? For the case $n=2$, 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 $f:X\to X$ is an expansive homeomorphism of a circle-like continuum $X$, then $f$ is itself weakly chaotic in the sense of Devaney.

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