Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I = [0, 1] to itself. For f ∈ C and x ∈ I, let ω(x,f ) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x, f ) to be the intersection of all the asymptotically stable sets of f containing ω(x,f ). We show that Q(x, f ) tells us quite a bit about the stability of ω(x,f ) subject to perturbations of either x or f , or both. For example, a chain recurrent point y is contained in Q(x, f ) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x, f ) ∈ I × C to Q(x, f ). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I × C. We also consider the map Qf : I →K given by x →Q(x, f ), and find that this map is continuous if and only if it is a constant map; that is, only when the set Q(f ) = {Q(x, f ): x ∈ I } is a singleton. © 2005 Elsevier Inc. All rights reserved