Devaney’s chaos on uniform limit maps

Kesong Yan, Fanping Zeng and Gengrong Zhang

Chaos, Solitons & Fractals, 2011, vol. 44, issue 7, 522-525

Abstract: Let (X,d) be a compact metric space and fn:X→X a sequence of continuous maps such that (fn) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all fn mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:44:y:2011:i:7:p:522-525

DOI: 10.1016/j.chaos.2011.05.006

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