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

Chris Good, ,School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK, Robert Leek, Joel Mitchell
2020 Discrete and Continuous Dynamical Systems. Series A  
We study sensitivity, 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
more » ... itive 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. The approach we take is topological rather than metric.
doi:10.3934/dcds.2020121 fatcat:3mz3hlq7andd5l7f4iq3uap5aa