Induced Dynamics of Non-Autonomous Discrete Dynamical Systems

TL;DR

This paper explores how the dynamical properties of non-autonomous discrete systems are reflected in their induced hyperspace systems, establishing conditions for property transfer and providing examples of non-extendable notions.

Contribution

It provides new conditions under which dynamical behaviors of non-autonomous systems are preserved or reflected in their hyperspace induced systems, and discusses property transfer and limitations.

Findings

Conditions for property extension between systems and hyperspaces.

Examples of non-extendable dynamical notions.

Analysis of properties like transitivity, mixing, entropy, and chaos.

Abstract

In this paper, we investigate the dynamics on the hyperspace induced by a non-autonomous dynamical system $(X,\mathbb{F})$, where the non-autonomous system is generated by a sequence $(f_n)$ of continuous self maps on $X$. We relate the dynamical behavior of the induced system on the hyperspace with the dynamical behavior of the original system $(X,\mathbb{F})$. We derive conditions under which the dynamical behavior of the non-autonomous system extends to its induced counterpart(and vice-versa). In the process, we discuss properties like transitivity, weak mixing, topological mixing, topological entropy and various forms of sensitivities. We also discuss properties like equicontinuity, dense periodicity and Li-Yorke chaoticity for the two systems. We also give examples when a dynamical notion of a system cannot be extended to its induced counterpart (and vice-versa).