Quotients of dynamical systems and chaos on the Cantor fan

TL;DR

This paper introduces a framework for analyzing quotient dynamical systems and applies it to study various notions of chaos on the Cantor fan and Lelek fan, revealing nuanced differences in chaotic behavior.

Contribution

It develops new results on the properties of quotient dynamical systems and applies them to distinguish different types of chaos on specific topological spaces.

Findings

Characterization of sensitive dependence in quotient systems

Conditions for transitivity in quotient systems

Existence of functions chaotic in some senses but not others on the Cantor fan

Abstract

Let $(X,f)$ be a dynamical system. Using an equivalence relation $\sim$ on $X$, we introduce the quotient $(X/_{\sim},f^{\star})$ of the dynamical system $(X,f)$. In the first part of the paper, we give new results about sensitive dependence on initial conditions of $(X/_{\sim},f^{\star})$, transitivity of $(X/_{\sim},f^{\star})$, and periodic points in $(X/_{\sim},f^{\star})$. In the second part of the paper, we use these results to study chaotic functions on the Cantor fan. Explicitly, we study functions $f$ on the Cantor fan $C$ such that (1) $(C,f)$ is chaotic in the sense of Devaney, (2) $(C,f)$ is chaotic in the sense of Robinson but not in the sense of Devaney, and (3) $(C,f)$ is chaotic in the sense of Knutzen but not in the sense of Devaney. We also study chaos on the Lelek fan.