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

DOI:
https://doi.org/10.1090/S0002-9947-97-01850-3

MathSciNet review:
1401776

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
54H20,
54F50,
54E50,
54B20

Retrieve articles in all journals with MSC (1991): 54H20, 54F50, 54E50, 54B20

Additional Information

**Hisao Kato**

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

Email:
hisakato@sakura.cc.tsukuba.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-97-01850-3

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