The space of ω-limit sets for Baire-1 functions on the interval

Abstract

Let I=[0,1] with bB1 the set of Baire-1 self-maps of I. There exists S a residual subset of bB1 such that for any f∈S, the following hold:1.The set of the ω-limit points Λ(f)=∪x∈Iω(x,f) is a nowhere dense and perfect subset of [0,1] with Hausdorff dimension zero.2.The collection of the ω-limit sets Ω(f)={ω(x,f):x∈I} generated by f is closed in the Hausdorff metric space.3.If x is a point at which f is continuous, then (x,f) is a point at which the map ω:I×bB1→K given by (x,f)→ω(x,f) is continuous.4.The n-fold iterate fn is an element of bB1 for all natural numbers n.