Chaotic behaviour of the map x ↦ ω(x, f)

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