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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Periodic orbits of continuous mappings of the circle


Author: Louis Block
Journal: Trans. Amer. Math. Soc. 260 (1980), 553-562
MSC: Primary 54H20; Secondary 58F20
DOI: https://doi.org/10.1090/S0002-9947-1980-0574798-8
MathSciNet review: 574798
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Abstract: Let f be a continuous map of the circle into itself and let $ P(f)$ denote the set of positive integers n such that f has a periodic point of period n. It is shown that if $ 1\, \in \,P(f)$ and $ n\, \in \,P(f)$ for some odd positive integer n then for every integer $ m\, > \,n$, $ m\, \in \,P(f)$. Furthermore, if $ P(f)$ is finite then there are integers m and n (with $ m\, \geqslant \,1$ and $ n\, \geqslant \,0$) such that $ P(f)\, = \,\{ m,\,2\,m,\,4\,m,\,8\,m,\,\ldots,\,{2^n}\,m\} $.


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

  • [1] L. Block, The periodic points of Morse-Smale endomorphisms of the circle, Trans. Amer. Math. Soc. 226 (1977), 77-88. MR 0436220 (55:9168)
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  • [3] A. N. Šarkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukrain 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/S0002-9947-1980-0574798-8
Article copyright: © Copyright 1980 American Mathematical Society

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