A characterization of ω-limit sets of maps of the interval with zero topological entropy

Abstract

Abstract We prove that an infinite W ⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iff W = Q ∪ P where Q is a Cantor set, and P is countable, disjoint from Q , dense in W if non-empty, and such that for any interval J contiguous to Q , card ( J ∩ P ) ≤ 1 if 0 or 1 is in J , and card ( J ∩ P ) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 that P can contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containing Q and contained in W , can be uncountable.