Periods of periodic points of maps of the circle which have a fixed point

Block, Louis

doi:10.1090/S0002-9939-1981-0612745-7

Periods of periodic points of maps of the circle which have a fixed point
HTML articles powered by AMS MathViewer

by Louis Block

Proc. Amer. Math. Soc. 82 (1981), 481-486

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

PDF | Request permission

Abstract:

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

References

Louis Block, Periodic orbits of continuous mappings of the circle, Trans. Amer. Math. Soc. 260 (1980), no. 2, 553–562. MR 574798, DOI 10.1090/S0002-9947-1980-0574798-8
Louis Block, John Guckenheimer, MichałMisiurewicz, 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
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
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