On tameness of almost automorphic dynamical systems for general groups

Fuhrmann, Gabriel; Kwietniak, Dominik

doi:10.1112/blms.12304

Cited by 10 publications

(21 citation statements)

References 33 publications

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“…Finally, we would like to mention that according to [Gla18, Corollary 5.4 (2)], minimal tame systems are always topo-isomorphic extensions of their maximal equicontinuous factor if the corresponding acting group is amenable (see also the short discussion at the end of Section 7.3). Systems belonging to this family are Sturmian-like Z n -actions [GM18a] or tame generalized Toeplitz shifts [FK20] (see also [LS18] for not necessarily tame but still mean equicontinuous examples). In fact, in [FK20] it is shown that every countable maximally almost periodic amenable group allows for effective mean equicontinuous minimal actions which are not equicontinuous (see also Section 7.3).…”

Section: Resultsmentioning

confidence: 99%

“…Systems belonging to this family are Sturmian-like Z n -actions [GM18a] or tame generalized Toeplitz shifts [FK20] (see also [LS18] for not necessarily tame but still mean equicontinuous examples). In fact, in [FK20] it is shown that every countable maximally almost periodic amenable group allows for effective mean equicontinuous minimal actions which are not equicontinuous (see also Section 7.3).…”

Section: Resultsmentioning

confidence: 99%

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The structure of mean equicontinuous group actions

2022

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We study mean equicontinuous actions of locally compact σ-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This characterization enables us to show the equivalence of mean equicontinuity and the weaker notion of Besicovitch-mean equicontinuity in fairly high generality, including actions of abelian groups as well as minimal actions of general groups. In the minimal case, we further conclude that mean equicontinuity is equivalent to discrete spectrum with continuous eigenfunctions. Applications of our results yield a new class of non-abelian mean equicontinuous examples as well as a characterization of those extensions of mean equicontinuous actions which are still mean equicontinuous.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

The structure of mean equicontinuous group actions

2022

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“…By the variational principle for topological entropy, the measuretheoretic entropy of a MPS equals the topological entropy of any of its topological models. However, this correspondence fails even for related notions like completely positive entropy [GW94], sequence entropy, and other notions of low complexity [KL07,FK20,FGJO18].…”

Section: Introductionmentioning

confidence: 99%

On topological models of zero entropy loosely Bernoulli systems

García-Ramos¹,

Kwietniak²

2020

Preprint

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We provide a purely topological characterisation of uniquely ergodic topological dynamical systems (TDSs) whose unique invariant measure is zero entropy loosely Bernoulli (following Ratner, we call such measures loosely Kronecker). At the heart of our proofs lies Feldman-Katok continuity (FK-continuity for short), that is, continuity with respect to the change of metric to the Feldman-Katok pseudometric. Feldman-Katok pseudometric is a topological analog of f-bar (edit) metric for symbolic systems. We also study an opposite of FK-continuity, coined FK-sensitivity. We obtain a version of Auslander-Yorke dichotomies: minimal TDSs are either FK-continuous or FK-sensitive, and transitive TDSs are either almost FKcontinuous or FK-sensitive.

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“…Of course there are many examples of AA systems which are not tame. For example there are almost automorphic systems having more than one invariant measure, or having positive entropy, and in [28,Theorem 3.11,Corollary 3.13] there are examples of regular AA systems which are not tame. Proposition 2.2.…”

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confidence: 99%

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

Glasner¹,

Megrelishvili²

2020

Preprint

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Todorcević' trichotomy in the class of separable Rosenthal compacta induces a hierarchy in the class of tame (compact, metrizable) dynamical systems (X, T ) according to the topological properties of their enveloping semigroups E(X). More precisely, we define the classeswhere Tame 1 is the proper subclass of tame systems with first countable E(X), and Tame 2 is its proper subclass consisting of systems with hereditarily separable E(X). We study some general properties of these classes and exhibit many examples to illustrate these properties.

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On tameness of almost automorphic dynamical systems for general groups

Cited by 10 publications

References 33 publications

The structure of mean equicontinuous group actions

The structure of mean equicontinuous group actions

On topological models of zero entropy loosely Bernoulli systems

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

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