Asymptotically stable sets and the stability of $\omega$-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 Q_f : 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.