Iterates of Borel functions

Abstract

Let B(α) be the set of bounded Borel-α self-maps of I=[0,1], where α is some countable ordinal. For f:I→I, let ω(x,f) be the ω-limit set generated by x∈I, and take CR(f) to be the set of chain recurrent points of f. There exists T a residual subset of B(α) such that for any f∈T, the following hold:1.The n-fold iterate fn is an element of B(α), for all natural numbers n.2.For any x∈I, the ω-limit set ω(x,f) is a Cantor set.3.For any ε>0, there exists a natural number M such that fm(I)⊂Bε(CR(f)), whenever m>M.4.The Hausdorff dimension dimH⁡CR(f)‾=0.5.There exists R, a residual subset of [0,1], with the property that ωf:R→K given by x⟼ω(x,f) is continuous.6.The function f is non-chaotic in the senses of Devaney and Li-Yorke.