Approximating ω-limit sets with periodic orbits

Summary.

Let f be a continuous self-map of I = [0, 1] with ω(x, f) the ω-limit set of f generated by x in I. We consider two closely related questions: when is $P(f) = \bigwedge(f)$, and more specifically, when is an ω-limit set contained in the Hausdorff closure of the periodic orbits? The connection between these properties of interval maps and their chaotic properties is investigated. In particular, we show that the property $\overline{P(f)} = \bigwedge(f)$ cannot be characterized in terms of the chaotic behavior of the function f, and that the condition $\overline{P(f)} = CR(f)$ insures that every ω-limit set of f is contained in the Hausdorff closure of the periodic orbits of f.