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

Christopher Mouron

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Displaying similar documents to “The solenoids are the only circle-like continua that admit expansive homeomorphisms”

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.

Whitney properties

J. Krasinkiewicz, Sam Nadler (1978)

Fundamenta Mathematicae

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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 which cannot be mapped onto any nonplaner circle-like continuum

George W. Henderson (1971)

Colloquium Mathematicae

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Whitney Preserving Maps onto Dendrites

Eiichi Matsuhashi (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X → D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D' with a dense set of branch points there exist a continuum X' containing a dense arc component and a Whitney preserving map f':X' → D' such that f' is not a homeomorphism.

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

Hisao Kato (1993)

Fundamenta Mathematicae

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We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions $W^{S} (x) | x \in X$ and $W^{(} u) (x) | x \in X$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, $W^{σ} (x)$ ...

The nonexistence of expansive homeomorphisms of dendroids

Hisao Kato (1990)

Fundamenta Mathematicae

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On the hyperspaces of snake-like and circle-like continua

J. Krasinkiewicz (1974)

Fundamenta Mathematicae

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Expansive homeomorphisms and indecomposability

Hisao Kuto (1991)

Fundamenta Mathematicae

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Chainable continua and homeomorphisms of the plane onto itself

Mirosław Sobolewski (1984)

Fundamenta Mathematicae

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Continua which admit only the identity mapping onto non-degenerate subcontinua

H. Cook (1967)

Fundamenta Mathematicae

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Exactly two-to-one maps from continua onto arc-continua

Wojciech Dębski, J. Heath, J. Mioduszewski (1996)

Fundamenta Mathematicae

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Continuing studies on 2-to-1 maps onto indecomposable continua having only arcs as proper non-degenerate subcontinua - called here arc-continua - we drop the hypothesis of tree-likeness, and we get some conditions on the arc-continuum image that force any 2-to-1 map to be a local homeomorphism. We show that any 2-to-1 map from a continuum onto a local Cantor bundle Y is either a local homeomorphism or a retraction if Y is orientable, and that it is a local homeomorphism if Y is not orientable. ...

Continuum many tent map inverse limits with homeomorphic postcritical ω-limit sets

Chris Good, Brian E. Raines (2006)

Fundamenta Mathematicae

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We demonstrate that the set of topologically distinct inverse limit spaces of tent maps with a Cantor set for its postcritical ω-limit set has cardinality of the continuum. The set of folding points (i.e. points at which the space is not homeomorphic to the product of a zero-dimensional set and an arc) of each of these spaces is also a Cantor set.

On composants of solenoids.

de Man, Ronald (1995)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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