Dynamics Of Finitely Generated Non-Autonomous Systems

TL;DR

This paper explores the relationship between the dynamics of non-autonomous systems generated by a finite family of functions and the composed autonomous system, analyzing properties like minimality and sensitivity.

Contribution

It establishes conditions linking the dynamical behaviors of non-autonomous and composed autonomous systems, with illustrative examples.

Findings

Conditions under which properties transfer between systems

Analysis of minimality, equicontinuity, and sensitivity

Examples demonstrating necessity of conditions

Abstract

In this paper, we discuss dynamical behavior of a non-autonomous system generated by a finite family $\mathbb{F}$. In the process, we relate the dynamical behavior of the non-autonomous system generated by the family $\mathbb{F}=\{f_1,f_2,\ldots,f_k\}$ with the dynamical behavior of the system $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$. We discuss properties like minimality, equicontinuity, proximality and various forms of sensitivities for the two systems. We derive conditions under which the dynamical behavior of $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$ is carried forward to $(X,\mathbb{F})$ (and vice-versa). We also give examples to illustrate the necessity of the conditions imposed.