Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

TL;DR

This paper introduces weaker forms of equicontinuity in topological and measure dynamical systems, exploring their links to discrete spectrum and zero sequence entropy, and providing new characterizations of these properties.

Contribution

It defines and analyzes weaker equicontinuity notions, establishing their relationships with discrete spectrum, sequence entropy, and sensitivity in both topological and measure-theoretic contexts.

Findings

Zero topological sequence entropy systems are strictly contained in diam-mean equicontinuous systems.

Transitive almost automorphic subshifts are diam-mean equicontinuous iff they are regular.

Discrete spectrum in ergodic systems is equivalent to {b5}-mean equicontinuity.

Abstract

We define weaker forms of topological and measure theoretical equicontinuity for topological dynamical systems and we study their relationships with systems with discrete spectrum and zero sequence entropy. In the topological category we show systems with zero topological sequence entropy are strictly contained in the diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is 1-1 on a set of full Haar measure). In the measure category we show that for ergodic topological systems having discrete spectrum is equivalent of being {\mu}-mean equicontinuous. For both categories we find characterizations using stronger versions of the classical notion of sensitivity.