Inheriting of chaos in nonautonomous dynamical systems

TL;DR

This paper investigates how chaos properties in a limiting map influence the chaotic behavior of nonautonomous discrete dynamical systems, showing inheritance of Li-Yorke and distributional chaos, and inheritance of infinite omega-limit sets when entropy is zero.

Contribution

It establishes that chaos in the limit map implies chaos in the nonautonomous system, extending understanding of chaos inheritance in nonautonomous dynamics.

Findings

Li-Yorke chaos is inherited from the limit map

Distributional chaos is also inherited

Infinite omega-limit sets are inherited when the limit map has zero topological entropy

Abstract

We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. We show, among others, that if $f$ is chaotic in the sense of Li and Yorke then the nonautonomous system $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $\omega$-limit sets.