TY - JOUR

T1 - An Estimate for the Number of Periodical Trajectories of the Given Period for Mapping of an Interval, Lucas Numbers, and Necklaces

AU - Ivanov, O. A.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Lucas numbers

KW - mappings of an interval

KW - number of necklaces

KW - periodical trajectory

KW - Sharkovskii’s ordering

UR - http://www.scopus.com/inward/record.url?scp=85061208443&partnerID=8YFLogxK

U2 - 10.3103/S1063454118040088

DO - 10.3103/S1063454118040088

M3 - Article

AN - SCOPUS:85061208443

VL - 51

SP - 367

EP - 372

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -