Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 367–372 | Cite as

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

  • O. A. IvanovEmail author


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 pq 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.


periodical trajectory mappings of an interval Sharkovskii’s ordering Lucas numbers number of necklaces 


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  1. 1.
    A. N. Sharkovskii “Coexistence of the cycles of a continuous mapping of the line into itself,” Ukr. Mat. Zh. 16, 61–71 (1964).Google Scholar
  2. 2.
    T.-Y. Li and J. A. Yorke “Period three implies chaos,” Am. Math. Mon. 82, 985–992 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. N. Sharkovskii S. F. Kolyada A. G. Sivak and V. V. Fedorenko Dynamics of One-Dimensional Mappings (Naukova Dumka, Kiev, 1989) [in Russian].Google Scholar
  4. 4.
    T. Lindström and H. Thunberg “An elementary approach to dynamics and bifurcation of skew-tent maps,” J. Differ. Equations Appl. 14, 819–833 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    O. Pfante and J. Jost Non-Generating Partition of Unimodular Maps, SFI Working Paper No. 2015–02–004 (Santa Fe Inst., Santa Fe, NM, 2015).Google Scholar
  6. 6.
    O. A. Ivanov Making Mathematics Come to Life: A Guide for Teachers and Students (MTsNMO, Moscow, 2009; Am. Math. Soc., Providence, RI, 2009).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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