scholarly journals Full groups of minimal homeomorphisms and Baire category methods

2014 ◽  
Vol 36 (2) ◽  
pp. 550-573
Author(s):  
TOMÁS IBARLUCÍA ◽  
JULIEN MELLERAY
Keyword(s):  
Baire Category ◽  
Full Group ◽  
Group Topology ◽  
Cantor Space ◽  
Polish Group ◽  
Minimal Homeomorphisms

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

scholarly journals ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM

2021 ◽  
pp. 1-31
Author(s):  
COLIN D. REID
Keyword(s):  
Minimal System ◽  
Full Group ◽  
Cantor Space

Abstract We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$ . We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$ .

scholarly journals Haar Null Sets and the Consistent Reflection of Non-meagreness

2014 ◽  
Vol 66 (2) ◽  
pp. 303-322 ◽  
Author(s):  
Márton Elekes ◽  
Juris Steprāns
Keyword(s):  
Probability Measure ◽  
Continuum Hypothesis ◽  
Borel Probability Measure ◽  
Baire Category ◽  
Borel Set ◽  
Polish Group ◽  
Borel Probability ◽  
Haar Null Sets ◽  
Definition Of ◽  
Null Set

AbstractA subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

scholarly journals Dynamical simplices and Fraïssé theory

2018 ◽  
Vol 39 (11) ◽  
pp. 3111-3126 ◽  
Author(s):  
JULIEN MELLERAY
Keyword(s):  
Invariant Measures ◽  
Probability Measures ◽  
Choquet Simplex ◽  
Cantor Space ◽  
Orbit Equivalent ◽  
Minimal Homeomorphisms

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$ . We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

On full groups of non-ergodic probability-measure-preserving equivalence relations

2015 ◽  
Vol 36 (7) ◽  
pp. 2218-2245 ◽  
Author(s):  
FRANÇOIS LE MAÎTRE
Keyword(s):  
Probability Measure ◽  
Operator Algebras ◽  
Baire Category ◽  
Equivalence Relations ◽  
Continuity Property ◽  
Algebraic Characterization ◽  
Full Group ◽  
Automatic Continuity ◽  
Ergodic Equivalence Relations ◽  
The Cost

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

scholarly journals A Tits alternative for topological full groups

2019 ◽  
Vol 41 (2) ◽  
pp. 622-640
Author(s):  
NÓRA GABRIELLA SZŐKE
Keyword(s):  
Free Group ◽  
Free Action ◽  
The Other ◽  
Full Group ◽  
Finitely Generated ◽  
Cantor Space ◽  
Cyclic Groups ◽  
Finitely Generated Groups ◽  
Tits Alternative ◽  
The One

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

scholarly journals Uniqueness of Polish group topology

2008 ◽  
Vol 155 (9) ◽  
pp. 992-999 ◽  
Author(s):  
Paul Gartside ◽  
Bojana Pejić
Keyword(s):  
Group Topology ◽  
Polish Group

$\operatorname{PL}(M)$ admits no Polish group topology

2017 ◽  
Vol 238 (3) ◽  
pp. 285-295
Author(s):  
Kathryn Mann
Keyword(s):  
Group Topology ◽  
Polish Group

scholarly journals Stable Analysis of Solution Set for System of Quasivariational Relations with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lefeng Shi ◽  
Zhe Yang
Keyword(s):  
Baire Category ◽  
Solution Set ◽  
Stability Of Solutions ◽  
Essential Component ◽  
Essential Stability ◽  
Essential Components ◽  
Quasivariational Inclusions

The essential stability of solutions for system of quasivariational relations is studied. We show that most of systems of quasivariational relations are essential (in the sense of Baire category) and that, for any system of quasivariational relations, there exists at least one essential component of its solution set. As applications, the existence of essential components of solution set for systems of KKM problems and systems of quasivariational inclusions is obtained.

Density and Baire category in recursive topology

2004 ◽  
Vol 50 (45) ◽  
pp. 381-391
Author(s):  
Iraj Kalantari ◽  
Larry Welch
Keyword(s):  
Baire Category ◽  
Recursive Topology
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