The universal minimal system for the group of homeomorphisms of the Cantor set

E. Glasner; B. Weiss

Displaying similar documents to “The universal minimal system for the group of homeomorphisms of the Cantor set”

On groups of essential values of topological cylinder cocycles over minimal rotations

Mieczysław K. Mentzen (2003)

Colloquium Mathematicae

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A theory of essential values of cocycles over minimal rotations with values in locally compact Abelian groups, especially $ℝ^{m}$ , is developed. Criteria for such a cocycle to be conservative are given. The group of essential values of a cocycle is described.

Cylinder cocycle extensions of minimal rotations on monothetic groups

Mieczysław K. Mentzen, Artur Siemaszko (2004)

Colloquium Mathematicae

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The main results of this paper are: 1. No topologically transitive cocycle $ℝ^{m}$ -extension of minimal rotation on the unit circle by a continuous real-valued bounded variation ℤ-cocycle admits minimal subsets. 2. A minimal rotation on a compact metric monothetic group does not admit a topologically transitive real-valued cocycle if and only if the group is finite.

Noninvertible minimal maps

Sergiĭ Kolyada, L&#039;ubomír Snoha, Sergeĭ Trofimchuk (2001)

Fundamenta Mathematicae

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For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{- 1} (A)$ share with A those topological properties which describe how large a set is. Using...

A note on universal minimal dynamical systems

Sławomir Turek (1991)

Commentationes Mathematicae Universitatis Carolinae

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Let $M (G)$ denote the phase space of the universal minimal dynamical system for a group $G$ . Our aim is to show that $M (G)$ is homeomorphic to the absolute of $D^{2^{ω}}$ , whenever $G$ is a countable Abelian group.

Induced subsystems associated to a Cantor minimal system

Heidi Dahl, Mats Molberg (2009)

Colloquium Mathematicae

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Let (X,T) be a Cantor minimal system and let (R,) be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system (Y,S) there exists a closed subset Z of X such that (Y,S) is conjugate to the subsystem (Z,T̃), where T̃ is the induced map on Z from T. We explore when we may choose Z to be a T-regular and/or a T-thin set, and we relate T-regularity of a set to R-étaleness. The latter concept plays an important role in the...

Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban, Lyubomyr Zdomskyy (2016)

Fundamenta Mathematicae

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We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property $α_{1.5}$ is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space $C_{p} (X)$ of continuous real-valued functions on X with the...

A study of remainders of topological groups

A. V. Arhangel'skii (2009)

Fundamenta Mathematicae

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Some duality theorems relating properties of topological groups to properties of their remainders are established. It is shown that no Dowker space can be a remainder of a topological group. Perfect normality of a remainder of a topological group is consistently equivalent to hereditary Lindelöfness of this remainder. No L-space can be a remainder of a non-locally compact topological group. Normality is equivalent to collectionwise normality for remainders of topological groups. If a...

Minimal and distal functions on semidirect products of groups

Paul Milnes (1988)

Colloquium Mathematicae

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Minimal actions of homeomorphism groups

Yonatan Gutman (2008)

Fundamenta Mathematicae

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Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ \subset 2^{2^{X}}$ , the space of maximal chains in $2^{X}$ , equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{H o m e o (X)}$ , the universal minimal space of Homeo(X), is not transitive...

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

S. Bezuglyi, K. Medynets (2008)

Colloquium Mathematicae

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We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ]. ...

Moscow spaces, Pestov-Tkačenko Problem, and $C$ -embeddings

Aleksander V. Arhangel'skii (2000)

Commentationes Mathematicae Universitatis Carolinae

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We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $μ G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $μ G$ . This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their $G_{δ}$ -modifications. We show, in particular, that the inequalities $in ({(H)}_{τ}) \leq 2^{τ \cdot in (H)}$ and $c ({(H)}_{τ}) \leq 2^{2^{τ \cdot in (H)}}$ hold for every topological group $H$ and every cardinal $τ \geq ω$ , where ${(H)}_{τ}$ denotes the underlying group $H$ endowed with the $G_{τ}$ -modification of the original topology of $H$ and $in (H)$ is the index of narrowness of the group $H$ . Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $ω$ -narrow group...

Equivalence of certain free topological groups

Jan Baars (1992)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we give a complete isomorphical classification of free topological groups $F M (X)$ of locally compact zero-dimensional separable metric spaces $X$ . From this classification we obtain for locally compact zero-dimensional separable metric spaces $X$ and $Y$ that the free topological groups $F M (X)$ and $F M (Y)$ are isomorphic if and only if $C_{p} (X)$ and $C_{p} (Y)$ are linearly homeomorphic.