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 of the circle to itself, let denote the set of positive integers such that has a periodic point of (least) period . Results are obtained which specify those sets, which occur as , for some continuous map 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.

**[1]**Louis Block,*Periodic orbits of continuous mappings of the circle*, Trans. Amer. Math. Soc.**260**(1980), no. 2, 553–562. MR**574798**, https://doi.org/10.1090/S0002-9947-1980-0574798-8**[2]**Louis Block, John Guckenheimer, Michał Misiurewicz, and Lai Sang Young,*Periodic points and topological entropy of one-dimensional maps*, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR**591173****[3]**O. M. Šarkovs′kiĭ,*Co-existence of cycles of a continuous mapping of the line into itself*, Ukrain. Mat. Ž.**16**(1964), 61–71 (Russian, with English summary). MR**0159905****[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), no. 3, 237–248. MR**0445556**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0612745-7

Article copyright:
© Copyright 1981
American Mathematical Society