Representations of Dynamical Systems on Banach Spaces

Glasner, Eli; Megrelishvili, Michael

doi:10.2991/978-94-6239-024-9_9

Cited by 27 publications

(77 citation statements)

References 111 publications

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“…In particular, representations on Banach spaces without a copy of l 1 (we call them Rosenthal Banach spaces) play a very important role in this hierarchy. According to the Rosenthal l 1 -dichotomy [36], and the corresponding Dynamical Bourgin-Fremlin-Talagrand dichotomy [13,15], there is a sharp dichotomy for metrizable dynamical systems; either their enveloping semigroup is of cardinality smaller or equal to that of the continuum, or it is very large and contains a copy of the Stone-Čech compactification of N (the set of natural numbers).…”

Section: Introductionmentioning

confidence: 99%

“…For compact metrizable systems X the two concepts discussed above coincide: (G, X) is tame iff it is Rosenthal representable. The case of metrizable X admits also interesting enveloping semigroup characterization: (G, X) is tame iff every element p ∈ E(G, X) is a Baire class 1 function p : X → X (see [19,15]). Recall that the enveloping semigroup E(G, X) of a compact G-system X is the pointwise closure of the set of all g-translations X → X (g ∈ G) in X X .…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.1. (See [29,13,15]) A representation of an action G X on a Banach space V is a pair (h, α), where h : G → Is(V ) is a strongly continuous cohomomorphism and α : X → V * is a weak-star continuous bounded map into the dual of V such that (h, α) respect the original action and the induced dual action. That is, v, α(gx) = h(g)(v), α(x) for all v ∈ V, g ∈ G, x ∈ X.…”

Section: Introductionmentioning

confidence: 99%

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Group actions on treelike compact spaces

Glasner

Megrelishvili

2019

Sci. China Math.

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We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler [37], 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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Group actions on treelike compact spaces

Glasner

Megrelishvili

2019

Sci. China Math.

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“…The last decade saw an increased interest in tame systems (see, for example, ; see also for an up to date account) revealing their connections to other areas of mathematics like Banach spaces , circularly ordered systems , substitutions and tilings, quasicrystals, cut and project schemes, and even model theory and logic . A major breakthrough in the general understanding of tameness was achieved by Glasner's recent structural result for tame minimal systems .…”

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confidence: 99%

On tameness of almost automorphic dynamical systems for general groups

Fuhrmann

Kwietniak

2019

Bull. London Math. Soc.

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Let (X,G) be a minimal equicontinuous dynamical system, where X is a compact metric space and G some topological group acting on X. Under very mild assumptions, we show that the class of regular almost automorphic extensions of (X,G) contains examples of tame but non‐null systems as well as non‐tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well‐studied families of group actions. These include Toeplitz flows over G‐odometers where G is countable and residually finite as well as symbolic extensions of irrational rotations.

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“…Such families play a major role in the theory of tame dynamical systems. See, for example, [12,11,14,15,16].…”

Section: Fragmented Mapsmentioning

confidence: 99%

A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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Abstract. 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 BVr (X, Y ) of all functions X → Y with variation ≤ 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 → Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.

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Representations of Dynamical Systems on Banach Spaces

Cited by 27 publications

References 111 publications

Group actions on treelike compact spaces

Group actions on treelike compact spaces

On tameness of almost automorphic dynamical systems for general groups

A note on tameness of families having bounded variation

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