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

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