An Estimate for the Number of Periodical Trajectories of the Given Period for Mapping of an Interval, Lucas Numbers, and Necklaces

Аннотация

In 1964, A.N. Sharkovskii 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 a closed bounded 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 3. Thus, if a mapping has a point of period 3, then it 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 the present paper, an exact lower bound for the number of trajectories of a given period for a mapping of a closed bounded interval into itself having a point of period 3 is given. The key point of the reasoning consisted in solution of a combinatorial problem the answer to which is expressed in terms of Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. We also consider a piecewise linear unimodular mapping of [0, 1] into itself for which it is possible to find points of an arbitrary given period.