For f: \(I\to I\), where \(I:=[0,1]\) and \(x\in I\), one defines \(f^ 0(x):=x\) and \(f^{n+1}:=f\circ f^ n\) and puts \(\gamma (x,f):=\{f^ n(x)\}^{\infty}_{n=0}.\) Then \(\omega\) (x,f), the \(\omega\)-limit set for f at x, is defined to be the cluster set of \(\gamma\) (x,f) and \(\Lambda (f):=\{\omega (x,f);\quad x\in I\}.\)
A function f: \(I\to I\) is Darboux (f\(\in {\mathcal D})\) if it maps intervals onto intervals. The main purpose of the article is to study the following question for several classes \({\mathcal E}\subset {\mathcal D}:\) What families of closed sets can constitute the family \(\Lambda\) (f) for a single \(f\in {\mathcal E}?\)
The authors obtain partial answers for \({\mathcal E}={\mathcal D}\cap {\mathcal B}_ i\), \(i=1,2\), and the complete answer for \({\mathcal E}={\mathcal D}\cap {\mathcal M}\). Here \({\mathcal B}_ i\) is the i-th Baire calss and \({\mathcal M}\) is the set of measurable functions. E.g., the following result is proved. Theorem. If Cl is any nonempty family of nonempty closed subsets of I, then \(Cl=\Lambda (f)\) for some \(f\in {\mathcal D}\cap {\mathcal M}\).