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Transactions of the American Mathematical Society
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
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\} $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0574798-8
PII: S 0002-9947(1980)0574798-8
Article copyright: © Copyright 1980 American Mathematical Society



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