Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

Hisao Kato

Displaying similar documents to “Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets”

On indecomposability and composants of chaotic continua

Hisao Kato (1996)

Fundamenta Mathematicae

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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 $d (f^{n} (x), f^{n} (y)) > c$ . A homeomorphism f: X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ ℤ such that $d i a m i f^{n} (A) > c$ . Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua...

The nonexistence of expansive homeomorphisms of dendroids

Hisao Kato (1990)

Fundamenta Mathematicae

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Continua with unique symmetric product

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Illanes (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a metric continuum. Let $F_{n} (X)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_{n} (X)$ provided that the following implication holds: if $Y$ is a continuum and $F_{n} (X)$ is homeomorphic to $F_{n} (Y)$ , then $X$ is homeomorphic to $Y$ . In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_{2} (X)$ , and (2) let $X$ be an arcwise...

The solenoids are the only circle-like continua that admit expansive homeomorphisms

Christopher Mouron (2009)

Fundamenta Mathematicae

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A homeomorphism h:X → X of a compactum X is expansive provided that for some fixed c > 0 and any distinct x, y ∈ X there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a circle-like continuum that admits an expansive homeomorphism, then X is homeomorphic to a solenoid.

Expansive homeomorphisms and indecomposability

Hisao Kuto (1991)

Fundamenta Mathematicae

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Hereditarily indecomposable continua with exactly n autohomeomorphisms

Elżbieta Pol (2002)

Colloquium Mathematicae

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The main goal of this paper is to construct, for every n,m ∈ ℕ, a hereditarily indecomposable continuum $X_{n m}$ of dimension m which has exactly n autohomeomorphisms.

Whitney properties

J. Krasinkiewicz, Sam Nadler (1978)

Fundamenta Mathematicae

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On hereditarily decomposable hereditarily equivalent non-metric continua

Lee Mohler, Lex Oversteegen (1990)

Fundamenta Mathematicae

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On span and chainable continua

Lex Oversteegen, E. Tymchatyn (1984)

Fundamenta Mathematicae

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The classification of circle-like continua that admit expansive homeomorphisms

Christopher Mouron (2011)

Fundamenta Mathematicae

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A homeomorphism h: X → X of a compactum X is expansive provided that for some fixed c > 0 and every x, y ∈ X (x ≠ y) there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a solenoid that admits an expansive homeomorphism, then X is homeomorphic to a regular solenoid. It can then be concluded that a circle-like continuum admits an expansive homeomorphism if and only if it is homeomorphic to a regular solenoid.

Non-planar embeddings of planar sets in $E^{3}$

D. McMillan (1974)

Fundamenta Mathematicae

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Monotone retractions and depth of continua

Janusz Jerzy Charatonik, Panayotis Spyrou (1994)

Archivum Mathematicum

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It is shown that for every two countable ordinals $α$ and $β$ with $α > β$ there exist $λ$ -dendroids $X$ and $Y$ whose depths are $α$ and $β$ respectively, and a monotone retraction from $X$ onto $Y$ . Moreover, the continua $X$ and $Y$ can be either both arclike or both fans.