A Descriptive Perspective on Devaney's Chaos and Some Results on Topologically Conjugate Systems

TL;DR

This paper revisits Devaney's chaos conditions using descriptive proximity, revealing that classical hierarchy results do not always apply and identifying invariance properties under topological conjugacy.

Contribution

It introduces a descriptive proximity framework for chaos, challenging classical hierarchy results and analyzing invariance under conjugacy.

Findings

Banks Theorem hierarchy does not hold in descriptive proximity

Descriptive transitivity and sensitivity are defined and analyzed

Some concepts remain invariant under topological conjugacy

Abstract

In this study, Devaney's chaos conditions are revisited within the framework of descriptive proximity. The concepts of descriptive transitivity, the density of descriptive periodic objects, and descriptive sensitivity are defined. The most notable finding of the study is that Banks Theorem, which establishes the hierarchy among these conditions in classical topology, does not generally hold in the descriptive perspective, and some of the concepts above remain invariant under topological conjugacy certain conditions.