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...

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.

Whitney properties

J. Krasinkiewicz, Sam Nadler (1978)

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.