A note on logarithmic mean equicontinuity

TL;DR

This paper introduces logarithmic mean equicontinuity in topological dynamical systems and shows it is equivalent to mean equicontinuity, providing a new perspective on ergodic properties and convergence behaviors.

Contribution

It characterizes unique ergodicity using logarithmic mean convergence and establishes the equivalence between logarithmic and mean equicontinuity.

Findings

Logarithmic mean equicontinuity is equivalent to mean equicontinuity.

A new characterization of unique ergodicity using harmonic limits.

Insights into convergence behaviors in topological dynamical systems.

Abstract

We study the set of harmonic limits of empirical measures in topological dynamical systems. We obtain a characterization of unique ergodicity based of logarithmic (harmonic) mean convergence in place of Ces\`aro convergence. We introduce logarithmic mean equicontinuity and show that a topological dynamical system is logarithmically mean equicontinuous if and only if it is mean equicontinuous.