The systems with almost Banach mean equicontinuity for abelian group actions

TL;DR

This paper introduces Banach-mean equicontinuity for abelian group actions and proves the equivalence of related concepts, showing that such systems have zero topological entropy, indicating low complexity.

Contribution

It establishes the equivalence of Banach-, Weyl-, and Besicovitch-mean equicontinuity for abelian group actions and links these to zero topological entropy.

Findings

Banach-, Weyl-, and Besicovitch-mean equicontinuity are equivalent for abelian group actions

Transitive, almost Banach-mean equicontinuous systems have zero topological entropy

Banach-mean equicontinuous systems under abelian group actions have zero topological entropy

Abstract

In this paper, we give the concept of Banach-mean equicontinuity and prove that three concepts, Bnanach-, Weyl- and Besicovitch-mean equicontinuity of a dynamic system with abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system with abelain group action is zero. As an application with our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an abelian groups is zero.