The structure of mean equicontinuous group actions

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.

Pure point spectrum for dynamical systems and mean, Besicovitch and Weyl almost periodicity

“…Hence, a point is Weyl almost periodic if and only if its orbit closure is mean equicontinuous. Thus, our results can be understood to provide a natural pointwise counterpart to the results of [16].…”

“…We use this for a strengthened version to be introduced in the following. Here we stick to the term ‘mean’ as this seems to be the common term within the study of equicontinuity properties in recent years (see, e.g., [11, 16, 19] as well as the discussion in Appendix C).…”

Amorphic complexity of group actions with applications to quasicrystals

“…To that end, we extend amorphic complexity to actions of locally compact, σ-compact and amenable groups. We will see that amorphic complexity is tailor-made to study strictly ergodic systems with discrete spectrum and continuous eigenfunctions, that is, minimal mean equicontinuous systems [FGL21,Corollary 1.6]. Most importantly, however, we show that amorphic complexity is not only theoretically well-behaved but also well-computable in specific examples.…”

“…We next discuss a natural class of dynamical systems with finite separation numbers: the class of mean equicontinuous systems, see [Aus59, Rob96, HJ97, Rob99, Cor06, Vor12, DG16, Gla18, LS18, FG20, FK20, GL20, FGL21] for numerous examples. In our discussion of mean equicontinuity, we follow the terminology of [FGL21]. Given a left or right Følner sequence F , a system pX, Gq is (Besicovitch) F -mean equicontinuous if for all ε ą 0 there is δ ą 0 such that for all x, y P X with dpx, yq ă δ we have…”

“…Recall that G acts effectively on X if for all g P G there is x P X such that gx ‰ x. According to [FGL21, Corollary 7.3], G is necessarily maximally almost periodic (see [FGL21] and references therein) if G allows for a minimal, mean equicontinuous and effective action on a compact metric space X. Hence, Lemma 3.5 gives that every minimal effective action by a group which is not maximally almost periodic (such as the continuous Heisenberg group H 3 pRq) is mean sensitive.…”