On definition of Devaney chaos for a continuous group action on a Hausdorff uniform space

TL;DR

This paper investigates Devaney chaos for continuous group actions on Hausdorff uniform spaces, clarifying the conditions under which chaos is characterized, especially regarding sensitivity and periodic points.

Contribution

It defines Devaney chaos for such systems, showing that dense periodic points and transitivity do not imply sensitivity, thus refining the chaos criteria.

Findings

Dense periodic points do not imply sensitivity in these systems.

Devaney chaos is characterized by transitivity, density of periodic points, and sensitivity.

The paper clarifies the conditions for chaos in group actions on uniform spaces.

Abstract

We show that the existence of a dense set of periodic points for a topologically transitive non-minimal continuous group action on a Hausdorff uniform space with an infinite acting group does not necessarily imply a sensitive dependence to the initial conditions in such a system. This leads to define the chaos in the sense of Devaney for a continuous group action on a Hausdorff uniform spaces with an infinite acting group in the original way, i.e. a non-minimal topologically transitive and sensitive system with a dense set of periodic points is a chaotic system in the sense of Devaney.