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

Abstract

Let K(2 N ) be the class of compact subsets of the Cantor space 2 N , furnished with the Hausdorff metric. Let ∈ C (2 N ). We study the map ω : 2 N → K(2 N ) defined as ω ( ) = ω( ), the ω-limit set of under . Unlike the case of -dimensional manifolds, ≥ 1, we show that ω is continuous for the generic self-map of the Cantor space, even though the set of functions for which ω is everywhere discontinuous on a subsystem is dense in C (2 N ). The relationships between the continuity of ω and some forms of chaos are investigated. MSC:37B99, 54H20, 37B10