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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Hisao Kato
Journal: Trans. Amer. Math. Soc. 349 (1997), 3645-3655
MSC (1991): Primary 54H20, 54F50; Secondary 54E50, 54B20
MathSciNet review: 1401776
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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.

References [Enhancements On Off] (What's this?)

  • [1] E. Akin, The General Topology of Dynamical Systems, Amer. Math. Soc., Providence, 1993. MR 94f:58041
  • [2] N. Aoki, Topological dynamics, in: Topics in general topology (eds, K. Morita and J. Nagata), North-Holland, Amsterdam, (1989), 625-740. MR 91m:58120
  • [3] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J., 15 (1948), 729-742. MR 10:261a
  • [4] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43-51. MR 13:265b
  • [5] C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math., 9 (1959), 653-659. MR 22:1867
  • [6] L. Fearnley, Characterizations of the continuous images of the pseudo-arc, Trans. Amer. Math. Soc., 111 (1964), 380-399. MR 29:596
  • [7] H. Kato, Expansive homeomorphisms in continuum theory, Topology Appl., 45 (1992), 223-243. MR 93j:54023
  • [8] -, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. MR 94k:54065
  • [9] -, Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke, Fund. Math., 145 (1994), 261-279. MR 95i:54049
  • [10] -, The nonexistence of expansive homeomorphisms of chainable continua, Fund. Math., 149 (1996), 119-126. CMP 96:09
  • [11] -, Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps of graphs, Pacific J. Math., 175 (1996), 93-116. CMP 97:04
  • [12] -, Minimal sets and chaos in the sense of Devaney on continuum-wise expansive homeomorphisms, Lecture Notes in Pure and Applied Mathematics, 170 (1995), 265-274. MR 96c:54065
  • [13] J. Kennedy, The construction of chaotic homeomorphisms on chainable continua, Topology Appl., 43 (1992), 91-116. MR 93b:54040
  • [14] A. Lelek, On weakly chainable continua, Fund. Math., 51 (1962), 271-282. MR 26:742
  • [15] W. Lewis, Most maps of the pseudo-arc are homeomorphisms, Proc. Amer. Math. Soc., 91 (1984), 147-154. MR 85g:54025
  • [16] W. Lewis, Continuous curves of pseudo-arcs, Houston J. Math., 11 (1985), 91-99. MR 86e:54038
  • [17] J. Mioduszewski, On a quasi-ordering in the class of continuous mappings of the closed interval onto itself, Colloq. Math., 9 (1962), 233-240. MR 26:741
  • [18] S. B. Nadler, Jr., Hyperspaces of sets, Pure and Appl. Math., 49 (Dekker, New York, 1978). MR 58:18330
  • [19] L. G. Oversteegen and E. D. Tymchatyn, On span and weakly chainable continua, Fund. Math., 122 (1984), 159-174. MR 85m:54034
  • [20] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. MR 12:344b
  • [21] R. F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc., 6 (1955), 308-309. MR 16:846d

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Additional Information

Hisao Kato
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan

Keywords: Expansive homeomorphism, decomposable, chainable, circle-like, the pseudo-arc, 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

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