On some properties of interval maps with zero topological entropy

Summary.

We consider the six properties of continuous maps of a compact interval: (i) f has zero topological entropy; (ii) Rec(f) is an F _σ-set; (iii) f is Lyapunov stable on Per(f); (iv) for any ε > 0, any infinite ω-limit set of f has a cover consisting of disjoint compact periodic intervals with length less than ε; (v) Per(f) is a G _δ set; (vi) every linearly ordered chain of ω-limit sets is countable. Some of these properties were basically studied in the sixties by A. N. Sharkovsky, and they were believed to be equivalent. But recently several authors have provided counterexamples. In this paper we complete these results, solve some open problems and disprove a recent conjecture. Thus, we show that (iv) ⇒ (iii) ⇒ (ii) ⇒ (i), (iv) ⇒ (vi) ⇒ (i), and (v) ⇒ (i), and that there is no other implication between these properties.