When is a dynamical system mean sensitive?

TL;DR

This paper investigates conditions under which topological dynamical systems are mean sensitive, providing new examples, counterexamples, and applications to sensitivity and equicontinuity properties.

Contribution

It establishes that uniquely ergodic, mixing systems with positive entropy are mean sensitive and introduces locally mean equicontinuous systems, addressing open questions in the field.

Findings

Uniquely ergodic, mixing systems with positive entropy are mean sensitive

Counterexample of a transitive system with positive entropy that is not mean sensitive

Mean sensitivity of the hyperspace does not imply mean sensitivity of the phase space

Abstract

This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.