arXiv subjects

Michael Megrelishvili

Publications and source records attributed to Michael Megrelishvili.

At least 37 records · Page 2Linked to original sources

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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arXiv2017

Tame functionals on Banach algebras

In the present note we introduce tame functionals on Banach algebras. A functional $f \in A^*$ on a Banach algebra $A$ is tame if the naturally defined linear operator $A \to A^*, a \mapsto f \cdot a$ factors through Rosenthal Banach spaces (i.e., not containing a copy of $l_1$). Replacing Rosenthal by reflexive we get a well known concept of weakly almost periodic functionals. So, always $WAP(A) \subseteq Tame(A)$. We show that tame functionals on the group algebra $l_1(G)$ are induced exactly by tame functions (in the sense of topological dynamics) on $G$ for every discrete group $G$. That is, $Tame(l_1(G))=Tame(G)$. Many interesting tame functions on groups come from dynamical systems theory. Recall that $WAP(L_1(G))=WAP(G)$ (Lau 1977, Ülger 1986) for every locally compact group $G$. It is an open question if $Tame(L_1(G))=Tame(G)$ holds for (nondiscrete) locally compact groups.

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arXiv2016

A note on tameness of families having bounded variation

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space $(Y,d)$ the compact space $BV_r(X,Y)$ of all functions $X \to Y$ with variation $\leq r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_+(X,Y)$ of all order preserving maps $X \to Y$ is sequentially compact where $Y$ is a compact metrizable partially ordered space in the sense of Nachbin.

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arXiv2016

Minimality of the Semidirect Product

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product $G\leftthreetimes P,$ where $G$ is a compact topological group and $P$ is a topological subgroup of $Aut(G)$. We prove that $G\leftthreetimes P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In case $G$ is abelian, the same is true for every subgroup $P \subseteq Aut(G)$. We show, in contrast, that there exist a compact two-step nilpotent group $G$ and a subgroup $P$ of $Aut(G)$ such that $G\leftthreetimes P$ is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik.

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arXiv2016

Eventual nonsensitivity and tame dynamical systems

In this paper we characterize tame dynamical systems and functions in terms of eventual non-sensitivity and eventual fragmentability. As a notable application we obtain a neat characterization of tame subshifts $X \subset \{0,1\}^{\mathbb Z}$: for every infinite subset $L \subseteq {\mathbb Z}$ there exists an infinite subset $K \subseteq L$ such that $π_{K}(X)$ is a countable subset of $\{0,1\}^K$. The notion of eventual fragmentability is one of the properties we encounter which indicate some "smallness" of a family. We investigate a "smallness hierarchy" for families of continuous functions on compact dynamical systems, and link the existence of a "small" family which separates points of a dynamical system $(G,X)$ to the representability of $X$ on "good" Banach spaces. For example, for metric dynamical systems the property of admitting a separating family which is eventually fragmented is equivalent to being tame. We give some sufficient conditions for coding functions to be tame and, among other applications, show that certain multidimensional analogues of Sturmian sequences are tame. We also show that linearly ordered dynamical systems are tame and discuss examples where some universal dynamical systems associated with certain Polish groups are tame.

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arXiv2016

Circularly ordered dynamical systems

We study topological properties of circularly ordered dynamical systems and prove that every such system is representable on a Rosenthal Banach space, hence, is also tame. We derive some consequences for topological groups. We show that several Sturmian like symbolic $\mathbb{Z}^k$-systems are circularly ordered. Using some old results we characterize circularly ordered minimal cascades.

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arXiv2016

Non-archimedean transportation problems and Kantorovich ultra-norms

We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.

math.FA↗

arXiv2015

Order and minimality of some topological groups

A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-open topology on $G$ is the smallest Hausdorff group topology on $G$. One of the key ideas is to verify that for such $X$ the Zariski and the Markov topologies on the group $H_+(X)$ coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn.

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arXiv2013

Banach representations and affine compactifications of dynamical systems

To every Banach space V we associate a compact right topological affine semigroup E(V). We show that a separable Banach space V is Asplund if and only if E(V) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of $l_1$) if and only if E(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

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arXiv2013

Free non-archimedean topological groups

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA groups frequently remain continuous. One of the main applications is: any epimorphism in the category NA must be dense. Moreover, the same methods improve the following result of T.H. Fay : the inclusion of a proper open subgroup H into G is not an epimorphism in the category of all Hausdorff topological groups. A key tool in the proofs is Pestov's test of epimorphisms. Our results provide a convenient way to produce surjectively universal NA abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao and Gao-Xuan as well as classical results about profinite groups which go back to Iwasawa and Gildenhuys-Lim.

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arXiv2011

Notes on non-archimedean topological groups

We show that the Heisenberg type group $H_X=(\Bbb{Z}_2 \oplus V) \leftthreetimes V^{\ast}$, with the discrete Boolean group $V:=C(X,\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characterization results for non-archimedean groups. Among others we show that every continuous group action of $G$ on a Stone space $X$ is a restriction of a continuous group action by automorphisms of $G$ on a topological (even, compact) group $K$. We show also that any epimorphism $f: H \to G$ (in the category of Hausdorff topological groups) into a non-archimedean group $G$ must be dense.

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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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arXiv2006

Compactifications of Semigroups and Semigroup Actions

An action of a topological semigroup S on X is compactifiable if this action is a restriction of a jointly continuous action of S on a Hausdorff compact space Y. A topological semigroup S is compactifiable if the left action of S on itself is compactifiable. It is well known that every Hausdorff topological group is compactifiable. This result cannot be extended to the class of Tychonoff topological monoids. At the same time, several natural constructions lead to compactifiable semigroups and actions. We prove that the semigroup C(K,K) of all continuous selfmaps on the Hilbert cube K is a universal second countable compactifiable semigroup (semigroup version of Uspenskij's theorem). Moreover, the Hilbert cube K under the action of C(K,K) is universal in the realm of all compactifiable S-flows X with compactifiable S where both X and S are second countable. We strengthen some related results of Kocak & Strauss and Ferry & Strauss about Samuel compactifications of semigroups. Some results concern compactifications with separately continuous actions, LMC-compactifications and LMC-functions introduced by Mitchell.

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arXiv2006

Every topological group is a group retract of a minimal group

We show that every Hausdorff topological group is a group retract of a minimal topological group. This first was conjectured by Pestov in 1983. Our main result leads to a solution of some problems of Arhangelskii. One of them is the problem about representability of a group as a quotient of a minimal group (Problem 519 in the first edition of "Open Problems in Topology"). Our approach is based on generalized Heisenberg groups and on groups arising from group representations on Banach spaces and in bilinear mappings.

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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

A note on sensitivity of semigroup actions

It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey, Silverman, and also by Glasner and Weiss. In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in a recent book of Glasner. Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.

math.DS↗

arXiv2006

Globalization of Confluent Partial Actions on Topological and Metric Spaces

We generalize Exel's notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe the universal induced global action on an extended set. This universal action can be lifted to the setting of topological spaces and continuous maps, as well as to that of metric spaces and non-expansive maps. Well-known constructions such as Shimrat's homogeneous extension are special cases of this construction. We investigate various properties of the arising spaces in relation to the original space; in particular, we prove embedding theorems and preservation properties concerning separation axioms and dimension. These results imply that every normal (metric) space can be embedded into a normal (metrically) ultrahomogeneous space of the same dimension and cardinality.

math.GN↗

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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