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

Chris Good; Robert Leek; Joel Mitchell

doi:10.3934/dcds.2020121

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2020, Volume 40, Issue 4: 2441-2474. Doi: 10.3934/dcds.2020121

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

School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

^* Corresponding author: jsm140@bham.ac.uk
^* Corresponding author: jsm140@bham.ac.uk

Received: July 2019

Revised: November 2019

Published: January 2020

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Abstract

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 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.

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Mathematics Subject Classification: Primary: 37B99, 54H20; Secondary: 37B10.

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Figure 1. A non-sensitive, eventually sensitive system

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