Abstract
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.
Highlights
Isometric actions on compact metric spaces constitute fundamental objects of study in the field of dynamical systems
We study mean equicontinuous actions of locally compact σ-compact amenable groups on compact metric spaces
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
Summary
Isometric actions on compact metric spaces constitute fundamental objects of study in the field of dynamical systems. Our results tie in with various recent streaks of investigations: for Z-actions, there is the fundamental work of Downarowicz and Glasner on mean equicontinuity [DG16], providing a detailed study in the minimal case Our results generalize these results from the group of integers to general locally compact σ-compact amenable groups. Concerning abelian groups, mean equicontinuity and its relation to the spectral theory of dynamical systems (in particular, to discrete spectrum) has been studied by various groups [GR17, Len, GRM19]. These works feature weaker versions of mean equicontinuity in order to characterize discrete. As discussed below (see Remark 6.4), one may argue that our spectral characterization shows that mean equicontinuous systems are the “right” systems to model minimal systems with aperiodic order
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