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The structure of mean equicontinuous group actions

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arxiv 1812.10219 v2 pith:2OLHHZDF submitted 2018-12-26 math.DS

Gabriel Fuhrmann , Maik Gröger , Daniel Lenz This is my paper · ORCID
classification math.DS
keywords meanequicontinuousactionsequicontinuitycharacterizationcompactgroupsequivalence
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abstract

We study mean equicontinuous actions of locally compact $\sigma$-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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Mean equicontinuity, almost automorphy and regularity

    math.DS 2019-08 conditional novelty 7.0 of 10

    For minimal systems, frequent stability is equivalent to the maximal equicontinuous factor being almost one-to-one, and diam-mean equicontinuity is equivalent to that factor being regular.

  2. The systems with almost Banach mean equicontinuity for abelian group actions

    math.DS 2019-09 conditional novelty 6.0 of 10

    For countable abelian group actions, Banach-, Weyl- and Besicovitch-mean equicontinuity are equivalent, and transitive almost Banach-mean equicontinuous systems have zero topological entropy.

  3. Discrete spectrum for amenable group actions

    math.DS 2019-08 conditional novelty 5.0 of 10

    Discrete spectrum, bounded measure complexity, and the two mean-equicontinuity notions coincide for invariant measures of countable amenable group actions, with the equicontinuity parts requiring tempered Følner sequences.

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