Eli Glasner
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View article: On the class of systems which are disjoint from every ergodic system
On the class of systems which are disjoint from every ergodic system Open
In this note we give a fairly direct proof of a recent theorem of Gorska, Lemanczyk and de la Rue which characterises the class of measure preserving transformations that are disjoint from every ergodic measure preserving transformation. O…
View article: The Unique stationary boundary property
The Unique stationary boundary property Open
Let $G$ be a locally compact group and $μ$ an admissible probability measure on $G$. Let $(B,ν)$ be the universal topological Poisson $μ$-boundary of $(G,μ)$ and $Π_s(G)$ the universal minimal strongly proximal $G$-flow. This note is inspi…
View article: Crossed products of dynamical systems; rigidity Vs. strong proximality
Crossed products of dynamical systems; rigidity Vs. strong proximality Open
Given a dynamical system $(X, Γ)$, the corresponding crossed product $C^*$-algebra $C(X)\rtimes_{r}Γ$ is called reflecting, when every intermediate $C^*$-algebra $C^*_r(Γ)<\mathcal{A} < C(X)\rtimes_{r}Γ$ is of the form $\mathcal{A}=C(Y)\rt…
View article: On intermediate factors of a product of disjoint systems
On intermediate factors of a product of disjoint systems Open
We consider an intermediate factor situation in two categories: probability measure preserving ergodic theory and compact topological dynamics. In the first we prove a master-key theorem and examine a wide range of applications. In the sec…
View article: Topological characteristic factors and nilsystems
Topological characteristic factors and nilsystems Open
We prove that the maximal infinite step pro-nilfactor X_\infty of a minimal dynamical system (X,T) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one-to-one modification of \pi:X \rightarrow …
View article: Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups
Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups Open
Let $Γ$ be a countable group and $(X, Γ)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(Γ)<\mathcal{A}
View article: A family of distal functions and multipliers for strict ergodicity
A family of distal functions and multipliers for strict ergodicity Open
We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\Z)$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of…
View article: More on tame dynamical systems
More on tame dynamical systems Open
In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and establ…
View article: Rigid topologies on groups
Rigid topologies on groups Open
Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.
View article: An ergodic system is dominant exactly when it has positive entropy
An ergodic system is dominant exactly when it has positive entropy Open
An ergodic dynamical system $\mathbf {X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic measure preserving transformations. Trans. Moscow Math. So…
View article: Distal strongly ergodic actions
Distal strongly ergodic actions Open
Let \eta be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F_3 on a compact metric space X , adm…
View article: On some generic classes of ergodic measure preserving transformations
On some generic classes of ergodic measure preserving transformations Open
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transfor…
View article: An ergodic system is dominant exactly when it has positive entropy
An ergodic system is dominant exactly when it has positive entropy Open
An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In …
View article: On a question of Kazhdan and Yom Din
On a question of Kazhdan and Yom Din Open
Studying a question of Kazhdan and Yom Din we first prove some new fixed point theorems for linear actions of a discrete countable group $G$ on a dual Banach space $V^*$, and then show that the answer to the question in full generality is …
View article: A family of distal functions and multipliers for strict ergodicity
A family of distal functions and multipliers for strict ergodicity Open
We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\mathbb{Z})$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{…
View article: Bernoulli disjointness
Bernoulli disjointness Open
International audience
View article: Irregular model sets and tame dynamics
Irregular model sets and tame dynamics Open
We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictl…
View article: A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system
A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system Open
We show the existence, over an arbitrary infinite ergodic -dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotient…
View article: Todorcević' trichotomy and a hierarchy in the class of tame dynamical systems
Todorcević' trichotomy and a hierarchy in the class of tame dynamical systems Open
Todorcević' trichotomy in the class of separable Rosenthal compacta induces a hierarchy in the class of tame (compact, metrizable) dynamical systems $(X,T)$ according to the topological properties of their enveloping semigroups $E(X)$. Mor…
View article: On some generic classes of ergodic measure preserving transformations
On some generic classes of ergodic measure preserving transformations Open
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transfor…
View article: Topological characteristic factors and nilsystems
Topological characteristic factors and nilsystems Open
We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $π:X \rightarr…
View article: On the virtual automorphism group of a minimal flow
On the virtual automorphism group of a minimal flow Open
We introduce the notions ‘virtual automorphism group’ of a minimal flow and ‘semiregular flow’ and investigate the relationship between the virtual and actual group of automorphisms.
View article: On weak rigidity and weakly mixing enveloping semigroups
On weak rigidity and weakly mixing enveloping semigroups Open
The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an int…
View article: Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems
Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems Open
The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\bf AP}^{[d]}$ for $d\in \N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=Δ$, the…
View article: On the disjointness property of groups and a conjecture of Furstenberg
On the disjointness property of groups and a conjecture of Furstenberg Open
In his seminal 1967 paper "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" Furstenberg introduced the notion of disjointness of dynamical systems, both topological and measure preserving. In this p…
View article: On universal minimal proximal flows of topological groups
On universal minimal proximal flows of topological groups Open
In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present …
View article: Circular orders, ultrahomogeneity and topological groups
Circular orders, ultrahomogeneity and topological groups Open
We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are g…
View article: Distal strongly ergodic actions
Distal strongly ergodic actions Open
Let $η$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting a…
View article: A minimal PI cascade with $2^{\mathfrak{c}}$ minimal ideals
A minimal PI cascade with $2^{\mathfrak{c}}$ minimal ideals Open
We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where ${\mathfrak{c}} ={2^{\aleph_0}}$) minimal left ideals is PI. Then we show the existence o…