Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited

Abstract

We discuss topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke Dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. We define what it means for a system to be eventually sensitive; we give a dichotomy for transitive dynamical systems in relation to eventual sensitivity. Along the way we define a property called splitting and discuss its relation to some existing notions of chaos.Date: March 2018.

On inverse shadowing

“…We refer to such a δ as an eventual-sensitivity constant. Clearly sensitivity implies eventual sensitivity but, as demonstrated in [14], the converse is not true. It is also easy to see that neither sensitivity nor eventual sensitivity can be held in conjunction with equicontinuity.…”