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


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.


Entropy Neural Network Statistical Physic Complex System Lower Bound 
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    Šarkovskii, A. N.: Coexistence of cycles of a continuous map of a line into itself. Ukr. Mat. Z.16, 61–71 (1964)Google Scholar
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    Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval. I.H.E.S. preprint, November 1975Google Scholar
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    Guckenheimer, J.: On the bifurcation of maps of the interval. Preprint, 1976Google Scholar
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    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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