On Dynamics Generated by a Uniformly Convergent Sequence of Maps

TL;DR

This paper investigates how the dynamics of a non-autonomous system generated by a sequence of maps converging uniformly to a limit relate to the dynamics of the limit map, establishing conditions for their properties to coincide.

Contribution

It provides new conditions under which dynamical properties of non-autonomous systems and their limit autonomous systems are equivalent, especially involving uniform convergence and feeble openness.

Findings

Equivalence of properties like equicontinuity, minimality, and proximality for the two systems.

Conditions under which transitivity, weak mixing, and sensitivities are equivalent.

Feeble openness ensures the equivalence of topological mixing and other mixing properties.

Abstract

In this paper, we study the dynamics of a non-autonomous dynamical system $(X,\mathbb{F})$ generated by a sequence $(f_n)$ of continuous self maps converging uniformly to $f$. We relate the dynamics of the non-autonomous system $(X,\mathbb{F})$ with the dynamics of $(X,f)$. We prove that if the family $\mathbb{F}$ commutes with $f$ and $(f_n)$ converges to $f$ at a "sufficiently fast rate", many of the dynamical properties for the systems $(X,\mathbb{F})$ and $(X,f)$ coincide. In the procees we establish equivalence of properties like equicontinuity, minimality and denseness of proximal pairs (cells) for the two systems. In addition, if $\mathbb{F}$ is feeble open, we establish equivalence of properties like transitivity, weak mixing and various forms of sensitivities. We prove that feeble openness of $\mathbb{F}$ is sufficient to establish equivalence of topological mixing for the two systems. We prove that if $\mathbb{F}$ is feeble open, dynamics of the non-autonomous system on a compact interval exhibits any form of mixing if and only if $(X,f)$ exhibits identical form of mixing. We also investigate dense periodicity for the two systems. We give examples to investigate sufficiency/necessity of the conditions imposed. In the process we derive weaker conditions under which the established dynamical relation (between the two systems $(X,\mathbb{F})$ and $(X,f)$) is preserved.