Hereditarily indecomposable continua with exactly n autohomeomorphisms

Elżbieta Pol

Displaying similar documents to “Hereditarily indecomposable continua with exactly n autohomeomorphisms”

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.

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...

Homeomorphisms of composants of Knaster continua

Sonja Štimac (2002)

Fundamenta Mathematicae

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The Knaster continuum $K_{p}$ is defined as the inverse limit of the pth degree tent map. On every composant of the Knaster continuum we introduce an order and we consider some special points of the composant. These are used to describe the structure of the composants. We then prove that, for any integer p ≥ 2, all composants of $K_{p}$ having no endpoints are homeomorphic. This generalizes Bandt’s result which concerns the case p = 2.

A note on the paper ``Smoothness and the property of Kelley''

Gerardo Acosta, Álgebra Aguilar-Martínez (2007)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, , Comment. Math. Univ. Carolin. (2000), no. 1, 123–132, it is claimed that $L (X) = ⋂_{p \in X} S (p)$ , where $L (X)$ is the set of points at which $X$ is locally connected and, for $p \in X$ , $a \in S (p)$ if and only if $X$ is smooth at $p$ with respect to $a$ . In this paper we show that such equality is incorrect and that the correct equality is $P (X) = ⋂_{p \in X} S (p)$ , where $P (X)$ is the set of points at which $X$ is connected im kleinen. We also use the correct equality to obtain some...

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 Suslinian number and other cardinal invariants of continua

T. Banakh, V. V. Fedorchuk, J. Nikiel, M. Tuncali (2010)

Fundamenta Mathematicae

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By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight...

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, $W^{σ} (x)$ ...

An Exactly Two-to-One Map from an Indecomposable Chainable Continuum

Jerzy Krzempek (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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It is shown that a certain indecomposable chainable continuum is the domain of an exactly two-to-one continuous map. This answers a question of Jo W. Heath.

Decompositions of the plane and the size of the continuum

Ramiro de la Vega (2009)

Fundamenta Mathematicae

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We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_{i}$ intersects each $E_{i}$ -class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition...

On the hyperspaces of hereditarily indecomposable continua

J. Krasinkiewicz (1974)

Fundamenta Mathematicae

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On a class of indecomposable continua

J. Krasinkiewicz (1974)

Colloquium Mathematicae

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Continua which admit no mean

K. Kawamura, E. Tymchatyn (1996)

Colloquium Mathematicae

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A symmetric, idempotent, continuous binary operation on a space is called a mean. In this paper, we provide a criterion for the non-existence of mean on a certain class of continua which includes tree-like continua. This generalizes a result of Bell and Watson. We also prove that any hereditarily indecomposable circle-like continuum admits no mean.