Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

TL;DR

This paper establishes a connection between mean equicontinuity and discrete spectrum in dynamical systems, introduces mean sensitivity and equicontinuity with respect to functions, and characterizes almost periodic functions in both topological and measure-theoretic contexts.

Contribution

It introduces the concepts of mean equicontinuity and mean sensitivity with respect to functions and characterizes almost periodic functions in measure-theoretic and topological settings.

Findings

Discrete spectrum characterized by mean equicontinuity.

Weakly almost periodic functions are mean equicontinuous.

Introduces new notions of mean sensitivity and equicontinuity.

Abstract

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold).