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Communications in Mathematical Physics

, Volume 54, Issue 3, pp 237–248 | Cite as

A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line

  • P. Štefan
Article

Abstract

Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.

Keywords

Entropy Neural Network Statistical Physic Complex System Lower Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Šarkovskii, A. N.: Coexistence of cycles of a continuous map of a line into itself. Ukr. Mat. Z.16, 61–71 (1964)Google Scholar
  2. 2.
    Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval. I.H.E.S. preprint, November 1975Google Scholar
  3. 3.
    Guckenheimer, J.: On the bifurcation of maps of the interval. Preprint, 1976Google Scholar
  4. 4.
    Li, T., Yorke, J. A.: Period three implies chaos. Am. Mat. Monthly82, 985–992 (1975)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • P. Štefan
    • 1
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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