A note on the set of periods for continuous maps of the circle which have degree one

Authors:
Lluís Alsedà and Jaume Llibre

Journal:
Proc. Amer. Math. Soc. **93** (1985), 133-138

MSC:
Primary 58F20; Secondary 54F62, 54H20

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766543-8

MathSciNet review:
766543

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Abstract: The main result of this paper is to complete Misiurewicz's characterization of the set of periods of a continuous map of the circle with degree one (which depends on the rotation interval of ). As a corollary we obtain a kind of perturbation theorem for maps of the circle of degree one, and a new algorithm to compute the set of periods when the rotation interval is known.

Also, for maps of degree one which have a fixed point, we describe the relationship between the characterizations of the set of periods of Misiurewicz and Block.

**[B1]**Louis Block,*Periods of periodic points of maps of the circle which have a fixed point*, Proc. Amer. Math. Soc.**82**(1981), no. 3, 481–486. MR**612745**, https://doi.org/10.1090/S0002-9939-1981-0612745-7**[BGMY]**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****[M]**Michał Misiurewicz,*Periodic points of maps of degree one of a circle*, Ergodic Theory Dynamical Systems**2**(1982), no. 2, 221–227 (1983). MR**693977****[NPT]**S. Newhouse, J. Palis, and F. Takens,*Bifurcations and stability of families of diffeomorphisms*, Inst. Hautes Études Sci. Publ. Math.**57**(1983), 5–71. MR**699057****[St]**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****[Sa]**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**

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0766543-8

Article copyright:
© Copyright 1985
American Mathematical Society