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Eli Glasner

Publications and source records attributed to Eli Glasner.

At least 37 records · Page 2Linked to original sources
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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arXiv2021

On two problems concerning topological centers

Let G be an infinite discrete group and bG its Cech-Stone compactification. Using the well known fact that a free ultrafilter on an infinite set is nonmeasurable, we show that for each element p of the remainder bG G, left multiplication L_p:bG \to bG is not Borel measurable. Next assume that G is abelian. Let D \subset \ell^\infty(G)$ denote the subalgebra of distal functions on G and let G^D denote the corresponding universal distal (right topological group) compactification of G. Our second result is that the topological center of G^D (i.e. the set of p in G^D for which L_p:G^D \to G^D is a continuous map) is the same as the algebraic center and that for G=Z (the group of integers) this center coincides with the canonical image of G in G^D.

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arXiv2020

Distal strongly ergodic actions

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 an invariant probability measure $μ$, such that the resulting dynamical system $(X, μ, F_3)$ is strongly ergodic and distal of rank $η$. In particular this shows that there is a $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.

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arXiv2020

Bernoulli disjointness

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $βG$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable $G$-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if $G$ is a countable icc group, then it admits a free, minimal, proximal flow.

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arXiv2020

Topological characteristic factors and nilsystems

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 $π:X \rightarrow X_\infty$, the induced open extension $π^*:X^* \rightarrow X^*_\infty$ has the following property: for $x$ in a dense $G_δ$ set of $X^*$, the orbit closure $L_x=\overline{\mathcal{O}}((x,x,\ldots,x), T\times T^2\times \ldots \times T^d)$ is $(π^*)^{(d)}$-saturated, i.e. $L_x=((π^*)^{(d)})^{-1}(π^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $k\ge 2$, then for any $d\in {\mathbb N}$ and any $0\le j<k$ there is a sequence $\{n_i\}$ of $\mathbb Z$ with $n_i\equiv j\ (\text{mod}\ k)$ such that $T^{n_i}x\rightarrow x, T^{2n_i}x\rightarrow x, \ldots, T^{dn_i}x\rightarrow x$ for $x$ in a dense $G_δ$ subset of $X$; (2) if $(X,T)$ is totally minimal, then $\{T^{n^2}x:n\in {\mathbb Z}\}$ is dense in $X$ for $x$ in a dense $G_δ$ subset of $X$; (3) for any $d\in\mathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${\bf RP}^{[d]}={\bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.

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arXiv2019

Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

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 ${\bf AP}^{[d]}=Δ$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. It follows that for a minimal $\infty$-pro-nilsystem, ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

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arXiv2019

Group actions on treelike compact spaces

We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler implies that every action of a topological group $G$ on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Helly's selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.

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arXiv2019

On the disjointness property of groups and a conjecture of Furstenberg

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 paper he showed that for actions of the integers the Bernoulli system $Ω= \{0, 1\}^\mathbb{Z}$, is disjoint from every minimal system, and that the subring $R_0$, over the field $\mathbb{Z}_2 =\{0, 1\}$, generated by the minimal functions in $Ω$, is a proper subset of $Ω$. He conjectured that a similar result holds in general and in our 1983 work "Interpolation sets for subalgebras of $l^\infty(\mathbb{Z})$" we confirmed this by showing that the closed subalgebra $\mathfrak{A}$ of $l^\infty(\mathbb{Z})$, generated by the minimal functions, is a proper subalgebra of $l^\infty(\mathbb{Z})$. In this work we generalize these results to a large class of groups. We call a countable group $G$ a DJ group if for every metrizable minimal action of $G$ there exists an essentially free minimal action disjoint from it. We show that amenable groups are DJ and that the DJ property is preserved under direct products. We define a simple dynamical condition DDJ on minimal systems, which is a strengthening of the Gottchalk-Hedlund property, and we say that a group $G$ is DDJ if every minimal $G$-system has this property. The DJ property implies DDJ and by means of an intricate construction we show that every finitely generated DDJ group is also DJ. Residually finite, maximally almost periodic and $C^*$-simple groups are all DDJ. Finally we show that Furstenberg's conjecture holds for every DDJ group.

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arXiv2018

On weak rigidity and weakly mixing enveloping semigroups

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 introductory section recalling some definitions and classic results, we establish some necessary conditions for this to happen, and in the final section we show, using Ratner's theory, that the enveloping semigroup of the `time one map' of a classical horocycle flow is weakly mixing.

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arXiv2018

Irregular model sets and tame dynamics

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 strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces. In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting.

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arXiv2018

Relative weak mixing is generic

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergodic extensions of a fixed, but arbitrary, ergodic transformation $T_0$. We then use a result of Connes, Feldman and Weiss to extend this relative theorem to the general (countable) amenable group.

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arXiv2018

Higher order regionally proximal equivalence relations for general minimal group actions

We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these relations are equivalence relations whenever the action is minimal. This was known for abelian actions by a result of Shao and Ye. We also show that these relations lift through extensions between minimal systems. Answering a question by Tao, given a minimal system, we prove that the regionally proximal equivalence relation of order $d$ corresponds to the maximal dynamical Antolín Camarena--Szegedy nilspace factor of order at most $d$. In particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor. Finally by using a result of Gutman, Manners and Varjú under some restrictions on the acting group, it follows that the regionally proximal equivalence relation of order $d$ corresponds to the maximal pronilfactor of order at most $d$ (a factor which is an inverse limit of nilsystems of order at most $d$).

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arXiv2018

A minimal PI cascade with $2^{\mathfrak{c}}$ minimal ideals

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 of various minimal PI flows with many minimal left ideals, as follows. For the acting group $G=SL_2(\mathbb{R})^\mathbb{N}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in \cite{GW-79} to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on and construct an example of a minimal PI-flow $(Y, \mathcal{G})$ on a compact manifold $Y$ and a suitable path-wise connected group $\mathcal{G}$ of homeomorphism of $Y$, such that the flow $(Y, \mathcal{G})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade $(X, T)$ which is PI (of order 3) and has $2^\mathfrak{c}$ minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication "less than $2^\mathfrak{c}$ minimal left ideals implies PI", fails.

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arXiv2018

On minimal actions of countable groups

Our purpose here is to review some recent developments in the theory of dynamical systems whose common theme is a link between minimal dynamical systems, certain Ramsey type combinatorial properties, and the Lovasz local lemma (LLL). For a general countable group G the two classes of minimal systems we will deal with are (I) the minimal subsystems of the {\em subgroup system} (Sub(G), G), called URS's (uniformly recurrent subgroups), and (II) minimal {\em subshifts}; i.e. subsystems of the binary Bernoulli G-shift ({0, 1}^G, {\sig_g}_{g \in G}).

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arXiv2017

On universal minimal proximal flows of topological groups

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 necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Theorem~II.4.4]) is strengthened. Finally, for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.

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arXiv2017

The structure of tame minimal dynamical systems for general groups

We use the structure theory of minimal dynamical systems to show that, for a general group $Γ$, a tame, metric, minimal dynamical system $(X, Γ)$ has the following structure: \begin{equation*} \xymatrix {& \tilde{X} \ar[dd]_π\ar[dl]_η& X^* \ar[l]_-{θ^*} \ar[d]^ι \ar@/^2pc/@{>}^{π^*}[dd]\\ X & & Z \ar[d]^σ\\ & Y & Y^* \ar[l]^θ} \end{equation*} Here (i) $\tilde{X}$ is a metric minimal and tame system (ii) $η$ is a strongly proximal extension, (iii) $Y$ is a strongly proximal system, (iv) $π$ is a point distal and RIM extension with unique section, (v) $θ$, $θ^*$ and $ι$ are almost one-to-one extensions, and (vi) $σ$ is an isometric extension. When the map $π$ is also open this diagram reduces to \begin{equation*} \xymatrix {& \tilde{X} \ar[dl]_η\ar[d]^ι \ar@/^2pc/@{>}^π[dd]\\ X & Z \ar[d]^σ\\ & Y } \end{equation*} In general the presence of the strongly proximal extension $η$ is unavoidable. If the system $(X, Γ)$ admits an invariant measure $μ$ then $Y$ is trivial and $X = \tilde{X}$ is an almost automorphic system; i.e. $X \oversetι{\to} Z$, where $ι$ is an almost one-to-one extension and $Z$ is equicontinuous. Moreover, $μ$ is unique and $ι$ is a measure theoretical isomorphism $ι: (X,μ, Γ) \to (Z, λ, Γ)$, with $λ$ the Haar measure on $Z$. Thus, this is always the case when $Γ$ is amenable.

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