Periodic orbits of continuous mappings of the circle
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- by Louis Block PDF
- Trans. Amer. Math. Soc. 260 (1980), 553-562 Request permission
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 $P(f) = \{ m, 2 m, 4 m, 8 m, \ldots , {2^n} m\}$.References
- Louis Block, The periodic points of Morse-Smale endomorphisms of the circle, Trans. Amer. Math. Soc. 226 (1977), 77–88. MR 436220, DOI 10.1090/S0002-9947-1977-0436220-1
- T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
- O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. . 16 (1964), 61–71 (Russian, with English summary). MR 0159905
- 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 445556
Additional Information
- © Copyright 1980 American Mathematical Society
- 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