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Periods of periodic points of maps of the circle which have a fixed point


Author: Louis Block
Journal: Proc. Amer. Math. Soc. 82 (1981), 481-486
MSC: Primary 58F20; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9939-1981-0612745-7
MathSciNet review: 612745
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Abstract: For a continuous map $ f$ of the circle to itself, let $ P(f)$ denote the set of positive integers $ n$ such that $ f$ has a periodic point of (least) period $ n$. Results are obtained which specify those sets, which occur as $ P(f)$, for some continuous map $ f$ of the circle to itself having a fixed point. These results extend a theorem of Šarkovskii on maps of the interval to maps of the circle which have a fixed point.


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, Global Theory of Dynamical Systems, Proceedings (Northwestern, 1979), Lecture Notes in Math., vol. 819, Springer-Verlag, Berlin and New York, 1980, pp. 18-34. MR 591173 (82j:58097)
  • [3] A. N. Šarkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukraïn. Mat. Ž. 16 (1964), 61-71. MR 0159905 (28:3121)
  • [4] 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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DOI: https://doi.org/10.1090/S0002-9939-1981-0612745-7
Article copyright: © Copyright 1981 American Mathematical Society

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