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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Minimal periodic orbits of continuous mappings of the circle


Author: Jaume Llibre
Journal: Proc. Amer. Math. Soc. 83 (1981), 625-628
MSC: Primary 54H20; Secondary 58F20
DOI: https://doi.org/10.1090/S0002-9939-1981-0627708-5
MathSciNet review: 627708
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Abstract: Let $ f$ be a continuous map of the circle into itself and suppose that $ n > 1$ is the least integer which occurs as a period of a periodic orbit of $ f$. Then we say that a periodic orbit $ \{ {p_1}, \ldots ,{p_n}\} $ is minimal if its period is $ n$. We classify the minimal periodic orbits, that is, we describe how the map $ f$ must act on the minimal periodic orbits. We show that there are $ \varphi (n)$ types of minimal periodic orbits of period $ n$, where $ \varphi $ is the Euler phi-function.


References [Enhancements On Off] (What's this?)

  • [1] L. Block, Periodic orbits of continuous mappings of the circle, Trans. Amer. Math. Soc. 260 (1980), 553-562. MR 574798 (83c:54057)
  • [2] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimensional maps, Proc. Conf. Global Theory of Dynamical Systems (Northwestern University) (Z. Nitecki and C. Robinson, eds.), Lecture Notes in Math., vol. 819, Springer-Verlag, Berlin and New York, 1980, pp. 18-34. MR 591173 (82j:58097)
  • [3] L. Block, Stability of periodic orbits in the theorem of Šarkovskii, Proc. Amer. Math. Soc. 81 (1981), 333-337. MR 593484 (82b:58071)
  • [4] R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill., 1971. MR 0283793 (44:1023)
  • [5] P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248. MR 0445556 (56:3894)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0627708-5
Article copyright: © Copyright 1981 American Mathematical Society

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