arXiv subjects

Eli Glasner

Publications and source records attributed to Eli Glasner.

69 records · Page 4Linked to original sources

arXiv2011

Translation-finite sets

The families of right (left) translation finite subsets of a discrete infinite group $Γ$ are defined and shown to be ideals. Their kernels $Z_R$ and $Z_L$ are identified as the closure of the set of products $pq$ ($p\cdot q$) in the Čech-Stone compactification $βΓ$. Consequently it is shown that the map $π: βΓ\to Γ^{WAP}$, the canonical semigroup homomorphism from $βΓ$ onto $Γ^{WAP}$, the universal semitopological semigroup compactification of $Γ$, is a homeomorphism on the complement of $Z_R \cup Z_L$.

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arXiv2011

The universal minimal space for groups of homeomorphisms of h-homogeneous spaces

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G=Homeo(X), is the space of maximal chains on X introduced by Uspenskij. If X is metrizable then clearly X is homeomorphic to the Cantor set and the result was already known (Glasner-Weiss 2003). However many new examples arise for non-metrizable spaces. These include, among others, the generalized Cantor sets X={0,1}^{kappa} for non-countable cardinals kappa, and the corona or remainder of omega, X=beta(omega) \ omega, where beta(omega) denotes the Stone-Cech compactification of the natural numbers.

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arXiv2010

On fixed point theorems and nonsensitivity

Sensitivity is a prominent aspect of chaotic behavior of a dynamical system. We study the relevance of nonsensitivity to fixed point theory in affine dynamical systems. We prove a fixed point theorem which extends Ryll-Nardzewski's theorem and some of its generalizations. Using the theory of hereditarily nonsensitive dynamical systems we establish left amenability of Asp(G), the algebra of Asplund functions on a topological group G (which contains the algebra WAP(G) of weakly almost periodic functions). We note that, in contrast to WAP(G), for some groups there are uncountably many invariant means on Asp(G). Finally we observe that dynamical systems in the larger class of tame G-systems need not admit an invariant probability measure.

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arXiv2010

On Hilbert dynamical systems

Returning to a classical question in Harmonic Analysis we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers Z which is not in the norm-closure of the algebra B(Z) of Fourier-Stieltjes transforms of measures on the circle, the dual group of Z, and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable.

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arXiv2009

Stationary dynamical systems

Following works of Furstenberg and Nevo and Zimmer we present an outline of a theory of stationary (or m-stationary) dynamical systems for a general acting group G equipped with a probability measure m. Our purpose is two-fold: First to suggest a more abstract line of development, including a simple structure theory. Second, to point out some interesting applications; one of these is a Szemeredi type theorem for SL(2,R).

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arXiv2009

On two recurrence problems

We review some aspects of recurrence in topological dynamics and focus on two open problems. The first is an old one concerning the relation between Poincare and Birkhoff recurrence; the second, due to Boshernitzan, is about moving recurrence. We provide a partial answer to a topological version of the moving recurrence problem.

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arXiv2009

The group $\Aut(μ)$ is Roelcke precompact

Following a similar result of Uspenskij on the unitary group of a separable Hilbert space we show that with respect to the lower (or Roelcke) uniform structure the Polish group $G= \Aut(μ)$, of automorphisms of an atomless standard Borel probability space $(X,μ)$, is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(μ)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, all coincide. Again following Uspenskij we also conclude that $G$ is totally minimal.

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arXiv2009

A topological lens for a measure-preserving system

We introduce a functor which associates to every measure preserving system (X,B,μ,T) a topological system (C_2(μ),\tilde{T}) defined on the space of 2-fold couplings of μ, called the topological lens of T. We show that often the topological lens "magnifies" the basic measure dynamical properties of T in terms of the corresponding topological properties of \tilde{T}. Some of our main results are as follows: (i) T is weakly mixing iff \tilde{T} is topologically transitive (iff it is topologically weakly mixing). (ii) T has zero entropy iff \tilde{T} has zero topological entropy, and T has positive entropy iff \tilde{T} has infinite topological entropy. (iii) For T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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arXiv2006

On metrizable enveloping semigroups

When a topological group $G$ acts on a compact space $X$, its enveloping semigroup $E(X)$ is the closure of the set of $g$-translations, $g\in G$, in the compact space $X^X$. Assume that $X$ is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) $X$ is hereditarily almost equicontinuous; (2) $X$ is hereditarily non-sensitive; (3) for any compatible metric $d$ on $X$ the metric $d_G(x,y):=\sup\{d(gx,gy): g\in G\}$ defines a separable topology on $X$; (4) the dynamical system $(G,X)$ admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup $E(X)$ is metrizable.

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arXiv2006

The structure of tame minimal dynamical systems

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $β\N$, or it is a "tame" topological space whose topology is determined by the convergence of sequences. In the latter case the dynamical system is called tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost 1-1 extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure preserving system is measure theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

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arXiv2006

Generically there is but one self homeomorphism of the Cantor set

We describe a self-homeomorphism $R$ of the Cantor set $X$ and then show that its conjugacy class in the Polish group $H(X)$ of all homeomorphisms of $X$ forms a dense $G_δ$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

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arXiv2006

Hereditarily non-sensitive dynamical systems and linear representations

For an arbitrary topological group G any compact G-dynamical system (G,X) can be linearly G-represented as a weak*-compact subset of a dual Banach space V*. As was shown by Megrelishvili (2003), the Banach space V can be chosen to be reflexive iff the metric system (G,X) is weakly almost periodic (WAP). In this paper we study the wider class of compact G-systems which can be linearly represented as a weak*-compact subset of a dual Banach space with the Radon-Nikodym property. We call such a system a Radon-Nikodym system (RN). One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily not sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of G-systems such as WAP and LE (locally equicontinuous). We show that the Glasner-Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. We also show that for symbolic systems the RN property is equivalent to having a countable phase space; and that any Z-dynamical system (f,X), where X is either the unit interval or the unit circle and f: X\to X is a homeomorphism, is an RN system. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup of a compact metrizable HNS system is a separable Rosenthal compact, hence of cardinality less than or equal c. We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorucević dichotomy concerning Rosenthal compacts.

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arXiv2006

On tame enveloping semigroups

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of the Stone-Cech compactification of the natural numbers, or it is a "tame" topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame iff it is equicontinuous (ii) for an abelian acting group a tame metric minimal system is PI (hence a weakly mixing minimal system is never tame), and (iii) a tame minimal cascade has zero topological entropy. We also show that for minimal distal-but-not-equicontinuous systems the canonical map from the enveloping operator semigroup onto the Ellis semigroup is never an isomorphism. This answers a long standing open question. We give a complete characterization of minimal systems whose enveloping semigroup is metrizable. In particular it follows that for abelian acting group such a system is equicontinuous.

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arXiv2002

The Cantor set of linear orders on N is the universal minimal S_\infty-system

Each topological group $G$ admits a unique universal minimal dynamical system $(M(G),G)$. When $G$ is a non-compact locally compact group the phase space $M(G)$ of this universal system is non-metrizable. There are however topological groups for which $M(G)$ is the trivial one point system (extremely amenable groups), as well as topological groups $G$ for which $M(G)$ is a metrizable space and for which there is an explicit description of the dynamical system $(M(G),G)$. One such group is the topological group $S_\infty$ of all permutations of the integers ${\mathbb Z}$, with the topology of pointwise convergence. We show that $(M(S_\infty),S_\infty)$ is a symbolic dynamical system (hence in particular $M(S_\infty)$ is a Cantor set), and give a full description of all its symbolic factors. Among other facts we show that $(M(G),G)$ (and hence also every minimal $S_\infty$) has the structure of a two-to-one group extension of proximal system and that it is uniquely ergodic.

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arXiv1996

A zero-one law for dynamical properties

For any countable group $Γ$ satisfying the ``weak Rohlin property'', and for any dynamical property, the set of $Γ$-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice $\integers^{\times{d}}$; indeed, all countable discrete amenable groups. For $Γ$ an arbitrary countable group, let $\actsp$ be the set of $Γ$-actions on the unit circle $Y$. We establish an Equivalence theorem by showing that a dynamical property is Baire/meager/residual in $\actsp$ if and only if it is Baire/meager/residual in the set of shift-invariant measures on the product space $Y^{\timesΓ}$.

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