The structure of \(\omega\)-limit sets for continuous functions (Q804724)

Let f: [0,1]\(\to [0,1]\) be a continuous function and \(f^ n:=f\circ f^{n-1}\) and \(f^ 0(x):=x.\) If \(\gamma (x,f):=\{f^ n(x)\}^{\infty}_{n=0}\) is the orbit of x under f, then the \(\omega\)- limit set (or attractor set) \(\omega\) (x,f) is defined to be the cluster set of \(\gamma\) (x,f). The main result is the Theorem: Let F be a nonempty set. Then F is an \(\omega\)-limit set for some f iff either F is nowhere dense, or it is a union of finitely many nondegenerate closed intervals.