Chaos in nonautonomous discrete dynamical systems

We consider nonautonomous discrete dynamical systems ( I , f 1 , ∞ ) given by sequences { f n } n ⩾ 1 of surjective continuous maps f n : 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 , f 1 , ∞ ) 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.