# Equicontinuity, Transitivity and Sensitivity: the Auslander-Yorke Dichotomy Revisited

## TL;DR

This paper revisits the Auslander-Yorke dichotomy in dynamical systems, exploring equicontinuity, transitivity, and sensitivity, and introduces new classifications and properties related to chaos and continuity.

## Contribution

It generalizes the Auslander-Yorke dichotomy for minimal systems and introduces the concept of eventual sensitivity and splitting in dynamical systems.

## Key findings

- Classification of transitive systems via equicontinuity pairs
- Existence of transitive systems with even continuity pairs but no equicontinuity points
- Dichotomy for transitive systems regarding eventual sensitivity

## 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.