# The nonexistence of expansive homeomorphisms of chainable continua

Fundamenta Mathematicae (1996)

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

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topKato, 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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- [9] H. Kato, Knaster-like chainable continua admit no expansive homeomorphisms, unpublished.
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- [11] J. Kennedy, The construction of chaotic homeomorphisms on chainable continua, Topology Appl. 43 (1992), 91-116. Zbl0756.54019
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- [13] W. Lewis, Most maps of the pseudo-arc are homeomorphisms, Proc. Amer. Math. Soc. 91 (1984), 147-154. Zbl0553.54018
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