Chaotic behaviour of the map x →ω(x; f)

摘要

Let K(2~N) be the class of compact subsets of the Cantor space 2N, furnished with the Hausdorff metric. Let f 2 C(2~N). We study the map ω_f : 2~N → K(2~N) defined as ω_f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω_f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω_f is everywhere discontinuous on a subsystem is dense in C(2~N). The relationships between the continuity of c_f and some forms of chaos are investigated.