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 denote the set of positive integers *n* such that *f* has a periodic point of period *n*. It is shown that if and for some odd positive integer *n* then for every integer , . Furthermore, if is finite then there are integers *m* and *n* (with and ) such that .

**[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)****[2]**T. Li and J. A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), 985-992. MR**0385028 (52:5898)****[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