Mean proximality and mean Li-Yorke chaos

TL;DR

This paper establishes conditions under which a topological dynamical system exhibits mean Li-Yorke chaos, linking mean sensitivity, mean proximal pairs, and unique ergodicity.

Contribution

It proves that mean sensitivity combined with a specific mean proximal pair implies mean Li-Yorke chaos, and characterizes mean proximal systems as uniquely ergodic with a singleton support.

Findings

Mean sensitivity plus a mean proximal pair implies mean Li-Yorke chaos.

Mean proximal systems are exactly those that are uniquely ergodic with a singleton measure support.

Characterization of mean proximality in topological dynamical systems.

Abstract

We prove that if a topological dynamical system is mean sensitive and contains a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li-Yorke chaotic (DC2 chaotic). On the other hand we show that a system is mean proximal if and only if it is uniquely ergodic and the unique measure is supported on one point.