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Eli Glasner

Publications and source records attributed to Eli Glasner.

At least 19 recordsLinked to original sources
arXiv2026

Joint Point-Distality and structure theory for commuting actions

Let $G$ and $H$ act commutatively and minimally on a compact metrizable space $X$. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of $\mathbb T^3$ whose canonical Furstenberg towers are different, showing that the meet formula can be strict.

math.DS↗
arXiv2026

On jointly transitive commuting minimal homeomorphisms

In analogy with ergodic theory, we introduce the notions of joint topological transitivity and joint minimality for commuting minimal homeomorphisms on a compact metric space $X$. We prove an analogue of the Berend--Bergelson theorem for the first notion and investigate to what extent this theorem holds for the second.

math.DS↗
arXiv2026

Affine Evaluation Maps, Dimension Groups, and Fair measures

Let $X$ be a Cantor space and let $Q\subset M_{fc}(X)$ be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated \emph{Affine Evaluation Map} (AEM), which assigns to each clopen set $A\subset X$ the affine function \[ \widehat A(μ)=μ(A),\qquad μ\in Q, \] and study the geometric subset condition obtained by comparing these evaluation functions pointwise on $Q$. For a good geometric AEM we construct the ordered group \[ G_Q=C(X,\mathbb Z)/N_Q, \qquad N_Q=\left\{f\in C(X,\mathbb Z): \int f\,dμ=0\ \text{for every }μ\in Q\right\}, \] and show that it is a simple dimension group whose normalized state space is canonically $Q$. We prove that its order interval $[0,u]$ is precisely the clopen scale and that \[ J_Q=N_Q, \] where $J_Q$ is generated by the elementary relations $\mathbf 1_A-\mathbf 1_B$ with $\widehat A=\widehat B$. We also show that the full stabilizer $\mathcal H_Q$ has invariant-measure simplex exactly $Q$. Using the clopen-scale property and the Herman--Putnam--Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism $T$ with $Q=M_T(X)$. For a Cantor minimal system we identify $G_Q$ with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If $Q=M_G(X)$, then $Q$ canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on $Q$ and which are specifically $\mathbb Z$-dynamical. In particular, $Q$ is good if and only if there exists a minimal homeomorphism $T$ of $X$ such that \[ M_T(X)=M_G(X). \]

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arXiv2025

On intermediate factors of a product of disjoint systems

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 second we treat the case when one of the systems is distal and then provide some counterexamples.

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arXiv2025

New algebras of functions on topological groups arising from G-spaces

For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. It contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, SUC(G) is trivial. We study the Roelcke algebra (= UC(G) = right and left uniformly continuous functions) and SUC compactifications of the groups S(N), of permutations of a countable set, and H(C), the group of homeomorphisms of the Cantor set. For the first group we show that WAP(G)=SUC(G)=UC(G) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that SUC(G) is properly contained in UC(G) and for this group UC(G) does not yield a right topological semigroup compactification. We introduce the notion of fixed point on a class P of flows (P-fpp) and study in particular groups which are SUC-amenable and groups with the SUC-fpp (SUC-extreme amenability). We show that every Polish group G with metrizable M(G) is SUC-amenable and if, in addition, M(G) is proximal, then G is SUC-extremely amenable.

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arXiv2024

Crossed products of dynamical systems; rigidity Vs. strong proximality

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 $\mathcal{A}=C(Y)\rtimes_{r}Γ$, corresponding to a dynamical factor $X \rightarrow Y$. It is called almost reflecting if $\mathbb{E}(\mathcal{A}) \subset \mathcal{A}$ for every such $\mathcal{A}$. These two notions coincide for groups admitting the approximation property (AP). Let $Γ$ be a non-elementary convergence group or a lattice in $\text{SL}_d(\mathbb{R})$ for some $d \ge 2$. We show that any uniformly rigid system $(X,Γ)$ is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups $Γ$ and any uniformly rigid system $(X,\mathcal{B},μ, Γ)$ the crossed product algebra $L^{\infty}(X,μ)\rtimesΓ$ is reflecting. An inclusion of algebras $\mathcal{A}\subset\mathcal{B}$ is called $\textit{minimal ambient}$ if there are no intermediate algebras. As a demonstration of our methods, we construct examples of minimal ambient inclusions with various interesting properties in the $C^*$ and the von Neumann settings.

math.OA↗
arXiv2024

The Unique stationary boundary property

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 inspired by a recent result of Hartman and Kalantar. We show that for a locally compact second countable group $G$ the following conditions are equivalent: (i) the measure $ν$ is the unique $μ$-stationary measure on $B$, (ii) $(B,G) \cong (Π_s(G),G)$.

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arXiv2024

On the class of systems which are disjoint from every ergodic system

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. Our proof works just as well for any countable acting group.

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arXiv2024

Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups

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 $\mathcal{A}$ $$C^{*}_{r}(Γ)<\mathcal{A}<C(X)\rtimes_rΓ,$$ which is not of the form $\mathcal{A} = C(Y) \rtimes_r Γ$, corresponding to a factor map $(X,Γ) \to (Y,Γ)$. Here $ C^{*}_{r} (Γ)$ and $C(X) \rtimes_r Γ$ are the reduced $C^*$-algebras of $Γ$ and $(X,Γ)$ respectively. Our main results are (1) For $Γ$, which is not $C^*$-simple, if $(X,Γ)$ admits a $Γ$-invariant probability measure, then such a sub-algebra always exists. (2) For $Γ= \mathbb{Z}$ and $(X, Γ)$ an irrational rotation of the circle $X = S^1$, we give a full description of all these non-crossed-product subalgebras.

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arXiv2023

Rigid topologies on groups

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.

math.GR↗
arXiv2023

More on tame dynamical systems

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 establish a neat characterization of tame subshifts. We also provide sufficient conditions which ensure that certain coding functions are tame. Finally we discuss examples where certain universal dynamical systems associated with some Polish groups are tame.

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arXiv2022

On a question of Kazhdan and Yom Din

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 in the negative.

math.FA↗
arXiv2022

Circular orders, ultra-homogeneous order structures and their automorphism groups

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 generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame. We show that for every circularly ultrahomogeneous action $G \curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a "circular" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).

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arXiv2021

An ergodic system is dominant exactly when it has positive entropy

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 this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.

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arXiv2021

On some generic classes of ergodic measure preserving transformations

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 transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.

math.DS↗
arXiv2021

Todorcević' trichotomy and a hierarchy in the class of tame dynamical systems

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 $E(X)$. More precisely, we define the classes $\mathrm{Tame}_\mathbf{2} \subset \mathrm{Tame}_\mathbf{1} \subset \mathrm{Tame},$ where $\mathrm{Tame}_\mathbf{1}$ is the proper subclass of tame systems with first countable $E(X)$, and $\mathrm{Tame}_\mathbf{2}$ is its proper subclass consisting of systems with hereditarily separable $E(X)$. We study some general properties of these classes and exhibit many examples to illustrate these properties.

math.DS↗
arXiv2021

A family of distal functions and multipliers for strict ergodicity

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 $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence in particular does not coincide with $\mathcal{W}$. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within $\mathcal{D}$ or in general, is necessarily a constant. An example of a metric, strictly ergodic, distal flow is constructed which admits a non-strictly ergodic $2$-fold minimal self-joining. It then follows that the enveloping group of this flow is not strictly ergodic (as a $T$-flow). Finally we show that the distal, strictly ergodic Heisenberg nil-flow is relatively disjoint over its largest equicontinuous factor from $|\mathcal{W}|$.

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