Sharkovskii’s ordering and evaluations of the number of periodic trajectories of given period of a self-map of an interval

Abstract

In 1964, A. N. Sharkovskii published an article in which he introduced a special ordering on the set of positive integers. His ordering has the property that if p⊳q and a continuous mapping of an interval into itself has a point of period p, then it has a point of period q. Since the least number with respect to this ordering is the number 3, it follows that if such a mapping has a point of period 3, then it has points of every period. The latter result was rediscovered in 1975 by Li and Yorke, who published it in their paper “Period three implies chaos”. Their work led to global recognition of Sharkovskii’s theorem, and since then a great number of papers related to the study of mappings of an interval have appeared. One area of research concerns estimates of the number of periodic trajectories a map satisfying the conditions of Sharkovskyii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he gave the exact lower bound for the number of periodic trajectories of a given period. The present article provides a new, significantly shorter and more natural proof of his result.