The nonexistence of expansive homeomorphisms of chainable continua

Hisao Kato

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 119-126
  • ISSN: 0016-2736

Abstract

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A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c . In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.

How to cite

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Kato, Hisao. "The nonexistence of expansive homeomorphisms of chainable continua." Fundamenta Mathematicae 149.2 (1996): 119-126. <http://eudml.org/doc/212111>.

@article{Kato1996,
abstract = {A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n(x),f^n(y)) > c$. In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.},
author = {Kato, Hisao},
journal = {Fundamenta Mathematicae},
keywords = {expansive homeomorphism; chainable continuum; pseudo-arc; hereditarily indecomposable continuum; hyperspace},
language = {eng},
number = {2},
pages = {119-126},
title = {The nonexistence of expansive homeomorphisms of chainable continua},
url = {http://eudml.org/doc/212111},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Kato, Hisao
TI - The nonexistence of expansive homeomorphisms of chainable continua
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 119
EP - 126
AB - A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n(x),f^n(y)) > c$. In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.
LA - eng
KW - expansive homeomorphism; chainable continuum; pseudo-arc; hereditarily indecomposable continuum; hyperspace
UR - http://eudml.org/doc/212111
ER -

References

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  2. [2] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. Zbl0043.16803
  3. [3] L. Fearnley, Characterizations of the continuous images of the pseudo-arc, Trans. Amer. Math. Soc. 111 (1964), 380-399. Zbl0132.18603
  4. [4] O. H. Hamilton, A fixed point theorem for the pseudo-arc and certain other metric continua, Proc. Amer. Math. Soc. 2 (1951), 173-174. Zbl0054.07003
  5. [5] H. Kato, Expansive homeomorphisms and indecomposability, Fund. Math. 139 (1991), 49-57. Zbl0823.54028
  6. [6] H. Kato, Expansive homeomorphisms in continuum theory, Topology Appl. 45 (1992), 223-243. Zbl0790.54048
  7. [7] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598. Zbl0797.54047
  8. [8] H. Kato, Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke, Fund. Math. 145 (1994), 261-279. Zbl0809.54033
  9. [9] H. Kato, Knaster-like chainable continua admit no expansive homeomorphisms, unpublished. 
  10. [10] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. Zbl0061.40107
  11. [11] J. Kennedy, The construction of chaotic homeomorphisms on chainable continua, Topology Appl. 43 (1992), 91-116. Zbl0756.54019
  12. [12] A. Lelek, On weakly chainable continua, Fund. Math. 51 (1962), 271-282. Zbl0111.35203
  13. [13] W. Lewis, Most maps of the pseudo-arc are homeomorphisms, Proc. Amer. Math. Soc. 91 (1984), 147-154. Zbl0553.54018
  14. [14] S. B. Nadler, Jr., Hyperspaces of Sets, Pure and Appl. Math. 49, Dekker, New York, 1978. 
  15. [15] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. Zbl0040.09903

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