Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hereditarily indecomposable compacta do not admit expansive homeomorphisms

Authors: Hisao Kato and Christopher G. Mouron
Journal: Proc. Amer. Math. Soc. 136 (2008), 3689-3696
MSC (2000): Primary 54H20, 54F50; Secondary 54E40
Published electronically: May 15, 2008
MathSciNet review: 2415056
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A homeomorphism $ h:X\longrightarrow X$ is expansive provided that for some fixed $ c>0$ and every $ x, y\in X$ there exists an integer $ n$, dependent only on $ x$ and $ y$, such that d$ (h^n(x),h^n(y))>c$. It is shown that if $ X$ is a hereditarily indecomposable compactum, then $ h$ cannot be expansive.

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

  • 1. M. L. Curtis and G. S. Young, A theorem on dimension, Proc. Amer. Math. Soc. 3 (1952), 159-161. MR 0046640 (13:764i)
  • 2. H. Kato, Continuum-wise expansive homeomorphisms, Can. J. Math. 45 (1993), no. 3, 576-598. MR 1222517 (94k:54065)
  • 3. R. Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. MR 534124 (80i:58032)
  • 4. C. Mouron, Expansive homeomorphisms and indecomposable subcontinua, Topology Appl. 126 (2002), no. 1-2, 13-28. MR 1934249 (2003h:54049)
  • 5. C. Mouron, Tree-like continua do not admit expansive homeomorphisms, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3409-3413. MR 1913021 (2003f:54076)
  • 6. L. Oversteegen and E. Tymchatyn, On hereditarily indecomposable compacta, Geometric and Algebraic Topology, 407-417, Banach Center Publ., 18, PWN, Warsaw, 1986. MR 925879 (88m:54044)
  • 7. R.V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys 39 (1974), 85-131. MR 771099 (86f:58106)
  • 8. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer, New York, 1982. MR 648108 (84e:28017)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54H20, 54F50, 54E40

Retrieve articles in all journals with MSC (2000): 54H20, 54F50, 54E40

Additional Information

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

Christopher G. Mouron
Affiliation: Department of Mathematics, The University of Alabama at Birmingham, Birmingham, Alabama 39254
Address at time of publication: Department of Mathematics and Computer Science, Rhodes College, Memphis, Tennessee 38112

Keywords: Expansive homeomorphism, hereditarily indecomposable continuum
Received by editor(s): January 18, 2007
Received by editor(s) in revised form: August 3, 2007
Published electronically: May 15, 2008
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society