A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line

Abstract

Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.