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]**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