An estimate for the number of periodical trajectories of the given period for a mapping of an interval, Lucas numbers, and necklaces

Abstract

In 1964, A. N. Sharkovsky published an article in which he introduced a special ordering on the set of positive integers. This ordering has the property that if p q and a mapping of an interval into itself has a point of period p, then it has a point of period q. The least number with respect to this ordering is the number 3. Thus, if a mapping has a point of period 3, then this mapping has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke, who published the paper “Period three implies chaos”. In that paper, an exact lower bound for the number of trajectories of a given period for a mapping of an interval into itself, which has a point of period 3 is given. The key point of the reasoning consisted in solution of one combinatorial problem, the answer to which is expressed in terms of the Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. The article also examined a particular piecewise linear unimodular mapping of an interval [0, 1] into itself for which it is possible to find points of an arbitrary given period.