On the structure of the ω-limit sets for continuous maps of the interval

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα ∈ I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].