Chaos in nonautonomous discrete dynamical systems

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We consider nonautonomous discrete dynamical systems (I,f1,∞) given by sequences {fn}n⩾1 of surjective continuous maps fn:I→I converging uniformly to a map f:I→I. Recently it was proved, among others, that generally there is no connection between chaotic behavior of (I,f1,∞) and chaotic behavior of the limit function f. We show that even the full Lebesgue measure of a distributionally scrambled set of the nonautonomous system does not guarantee the existence of distributional chaos of the limit map and conversely, that there is a nonautonomous system with arbitrarily small distributionally scrambled set that converges to a map distributionally chaotic a.e.