Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical Systems

TL;DR

This paper extends the classical Auslander-Yorke dichotomy to non-autonomous discrete dynamical systems on uniform and topological spaces, establishing conditions under which systems are either sensitive or equicontinuous.

Contribution

It generalizes the Auslander-Yorke dichotomy to non-autonomous systems in uniform and T3 spaces, introducing new variants like syndetic equicontinuity and Hausdorff sensitivity.

Findings

Minimal periodic systems are either sensitive or equicontinuous.

Established dichotomies involving syndetic equicontinuity and thick sensitivity.

Proved analogues in T3 topological spaces.

Abstract

We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander-Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander-Yorke-type dichotomies on T3 spaces.