On Dynamics Generated by a Uniformly Convergent Sequence of Maps

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.