On Hilbert dynamical systems

Glasner, Eli; Weiss, Benjamin

doi:10.1017/s0143385711000277

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…For the other, if f ∈ Hilb(G), there is a sequence f n → f with f n ∈ B(G), and hence f factors through the compactification associated to the closed algebra A generated by the functions f n , which is metrizable (since A is separable) and Hilbert-representable (by the proposition). This question has also been investigated in [GW12]. In Section 4, we will see that the answer is positive for pro-oligomorphic groups.…”

Section: 1mentioning

confidence: 94%

“…This question has also been investigated in [GW12]. In Section 4, we will see that the answer is positive for pro-oligomorphic groups.…”

Section: Introductionmentioning

confidence: 94%

“…Given a function f ∈ RUC(G), let X f be the compactification of G associated to the leftinvariant closed subalgebra of RUC(G) generated by f . In [GW12,§2], it is observed that X f is Hilbert-representable whenever f is positive definite, in the case G = Z; more generally, the following holds.…”

Section: 1mentioning

confidence: 99%

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Eberlein oligomorphic groups

Yaacov

Ibarlucía

Tsankov

2017

Trans. Amer. Math. Soc.

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We study the Fourier-Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier-Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of ℵ 0 -stable, ℵ 0categorical structures. This analysis is then extended to all semitopological semigroup compactifica-2010 Mathematics Subject Classification. Primary: 22F50. Secondary: 03C45, 22A25, 03C98, 22A15.

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Section: 1mentioning

confidence: 94%

“…This question has also been investigated in [GW12]. In Section 4, we will see that the answer is positive for pro-oligomorphic groups.…”

Section: Introductionmentioning

confidence: 94%

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Eberlein oligomorphic groups

Yaacov

Ibarlucía

Tsankov

2017

Trans. Amer. Math. Soc.

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“…Surprisingly it is not known whether X f := cls {f • T n : n ∈ Z} ⊂ R Z is a Hilbert system when f : Z → R is a Hilbert function (see [39]). The following closely related question from [58] is also open: Is it true that Hilbert representable compact metric Z-spaces are closed under factors?…”

Section: A Characterization Of Tame Symbolic Systemsmentioning

confidence: 99%

More on tame dynamical systems

Glasner¹,

Megrelishvili²

2023

Preprint

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In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and establish a neat characterization of tame subshifts. We also provide sufficient conditions which ensure that certain coding functions are tame. Finally we discuss examples where certain universal dynamical systems associated with some Polish groups are tame.

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“…Surprisingly it is not known whether X f := cls {f • T n : n ∈ Z} ⊂ R Z is a Hilbert system when f : Z → R is a Hilbert function (see [44]). The following closely related question from [68] is also open: Is it true that Hilbert representable compact metric Z-spaces are closed under factors?…”

mentioning

confidence: 99%

Eventual nonsensitivity and tame dynamical systems

Glasner,

Megrelishvili

2014

Preprint

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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 ⊂ {0, 1} Z : for every infinite subset L ⊆ Z there exists an infinite subset K ⊆ 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 universal dynamical systems associated with certain Polish groups are tame. Contents 23 8. Order preserving systems are tame 31 9. Intrinsically tame groups 38 10. Appendix 41 References 43

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On Hilbert dynamical systems

Cited by 6 publications

References 20 publications

Eberlein oligomorphic groups

Eberlein oligomorphic groups

More on tame dynamical systems

Eventual nonsensitivity and tame dynamical systems

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