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

Glasner, Eli; Megrelishvili, Michael

doi:10.48550/arxiv.2011.04376

Cited by 2 publications

(6 citation statements)

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“…If K is metrizable then it is equivalent to say that the enveloping semigroup E(K) is "small"; namely, a separable Rosenthal compact space (see [15,16]). In view of a hierarchy of tame metric dynamical systems (see [19]) induced by the Todorcević' Trichotomy for Rosenthal compact spaces, we ask the following Question 1.4. Which (c-)orderly topological groups G admit an effective (c-)ordered action on a compact metrizable space K such that the enveloping semigroup E(K) is: a) metrizable?…”

Section: Do Not Know If the Circular Analog Of Theorem B Remains Truementioning

confidence: 99%

“…Deep results of Todorcević [32] and Argyros-Dodos-Kanellopoulos [1, Section 4.3.7]) about separable Rosenthal compacta, lead to a hierarchy of tame metric dynamical systems (see [19]) according to topological properties of corresponding enveloping semigroups. In view of this hierarchy we ask the following Question 4.10.…”

Section: 1mentioning

confidence: 99%

“…Ordered dynamical systems were studied in several recent publications concerning tame systems, Sturmian like circularly ordered symbolic systems and some dynamical generalizations of the amenability concept. See joint works with Eli Glasner [16,17,18,19] and also [27]. Investigation of order-preserving compact dynamical systems provides natural framework to study orderable groups (and the orderability itself).…”

Section: Introductionmentioning

confidence: 99%

“…The second reason comes from our papers [16,17,19] with Eli Glasner, where we study Sturmian like and other circularly ordered dynamical systems. In many such cases, the enveloping semigroup is a circularly ordered semigroup.…”

Section: Introductionmentioning

confidence: 99%

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Orderable groups and semigroup compactifications

Megrelishvili¹

2021

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Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups. Contents 1. Introduction 1 2. Linearly ordered groups 3 3. Circular order and its topology 9 4. Circularly ordered groups 15 5. Representations on Banach spaces 19 References 19

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Section: Do Not Know If the Circular Analog Of Theorem B Remains Truementioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Orderable groups and semigroup compactifications

Megrelishvili¹

2021

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“…A Banach space V does not contain an l 1 -sequence (equivalently, does not contain an isomorphic copy of l 1 ) if and only if every bounded sequence in V has a weak-Cauchy subsequence [27]. As in [8,9,10], we call a Banach space satisfying these equivalent conditions a Rosenthal space. Every reflexive space is Asplund and every Asplund is Rosenthal.…”

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confidence: 99%

Topological group actions by group automorphisms and Banach representations

Megrelishvili¹

2021

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We study Banach representability for actions of topological groups on groups by automorphisms (in particular, actions of groups on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces.In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 (R). We show that the linear action of GLn(R) on R n , for every n ≥ 2, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 ).The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for SL 2 (R). As a byproduct we illuminate some counterexamples about Banach representations of homogeneous G-actions G/H.

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

Cited by 2 publications

References 40 publications

Orderable groups and semigroup compactifications

Orderable groups and semigroup compactifications

Topological group actions by group automorphisms and Banach representations

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