Minimal periodic orbits of continuous mappings of the circle

Author:
Jaume Llibre

Journal:
Proc. Amer. Math. Soc. **83** (1981), 625-628

MSC:
Primary 54H20; Secondary 58F20

DOI:
https://doi.org/10.1090/S0002-9939-1981-0627708-5

MathSciNet review:
627708

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Abstract: Let be a continuous map of the circle into itself and suppose that is the least integer which occurs as a period of a periodic orbit of . Then we say that a periodic orbit is minimal if its period is . We classify the minimal periodic orbits, that is, we describe how the map must act on the minimal periodic orbits. We show that there are types of minimal periodic orbits of period , where is the Euler phi-function.

**[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]**Louis Block,*Stability of periodic orbits in the theorem of Šarkovskii*, Proc. Amer. Math. Soc.**81**(1981), no. 2, 333–336. MR**593484**, https://doi.org/10.1090/S0002-9939-1981-0593484-8**[4]**Robert F. Brown,*The Lefschetz fixed point theorem*, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR**0283793****[5]**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/S0002-9939-1981-0627708-5

Article copyright:
© Copyright 1981
American Mathematical Society