Devaney's chaos and eventual sensitivity

TL;DR

This paper redefines Devaney's chaos for semiflows using eventual sensitivity, extends the Auslander-Yorke dichotomy to non-compact spaces, and explores the relationship between Devaney chaos and topological sensitivity.

Contribution

It introduces an equivalent characterization of Devaney chaos via eventual sensitivity and generalizes the dichotomy to broader phase spaces.

Findings

Devaney chaos can be characterized by eventual sensitivity.

The Auslander-Yorke dichotomy holds for non-compact phase spaces.

Open questions about the link between Devaney chaos and topological sensitivity.

Abstract

We give an equivalent definition of Devaney chaotic semiflow in terms of eventual sensitivity, the notion recently introduced by C.~Good, R.~Leek, and J.~Mitchell. As a consequence, we prove a version of Auslander-Yorke dichotomy for the case of not necessarily compact phase spaces. Finally, we raise some questions about the relation between Devaney chaoticity and the notion of topological sensitivity, which was recently introduced by C.~Good and C.~Mac\'ias.