Dynamics of Baire-2 functions on the interval

Abstract

Let I=[0,1], with bB2 the set of bounded Baire-2 self-maps of I. The set bB2 is closely related to the collection of measurable functions, as g:I→I is measurable if and only if g=f a.e., for some f∈bB2. For f∈bB2, let ω(x,f) be the ω-limit set generated by x∈I, and take Λ(f)=∪x∈Iω(x,f) to be the set of ω-limit points of f. There exists T a residual subset of bB2 such that for any f∈T, the following hold:1.For any x∈I, the ω-limit set ω(x,f) is a Cantor set.2.For any ε>0, there exists a natural number M such that fm(I)⊂Bε(Λ(f)), whenever m>M.3.The Hausdorff dimension dimH⁡Λ(f)‾=0.4.There exists R, a residual subset of [0,1], with the property that ωf:R→K given by x⟼ω(x,f) is continuous.5.The n-fold iterate fn is an element of bB2, for all natural numbers n.6.The function f is non-chaotic in the senses of Devaney and Li-Yorke.