Periodic points and topological entropy of maps of the circle

Author:
Chris Bernhardt

Journal:
Proc. Amer. Math. Soc. **87** (1983), 516-518

MSC:
Primary 58F20; Secondary 28D20, 54C70

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684649-7

MathSciNet review:
684649

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a continuous map from the circle to itself, let denote the set of integers for which has a periodic point of period . In this paper it is shown that the two smallest numbers in are either coprime or one is twice the other.

**[1]**C. Bernhardt,*Periodic orbits of continuous mappings of the circle without fixed points*, Ergodic Theory and Dynamical Systems**1**(1981), 413-417. MR**662734 (84h:58118)****[2]**L. Block,*Periods of periodic points of maps of the circle which have a fixed point*, Proc. Amer. Math. Soc.**82**(1981), 481-486. MR**612745 (82h:58042)****[3]**L. Block, E. M. Coven and Z. Nitecki,*Minimizing topological entropy for maps of the circle*, Ergodic Theory and Dynamical Systems**1**(1981), 145-149. MR**661815 (83h:58058)****[4]**L. Block, J. Guckenheimer, M. Misurewicz and L. Young,*Periodic points and topological entropy of one dimensional maps*, Global Theory of Dynamical Systems, Lecture Notes in Math., vol. 819, Springer-Verlag, Berlin and New York, 1980. MR**591173 (82j:58097)****[5]**C. Hardy and E. Wright,*An introduction to the theory of numbers*, Clarendon Press, Oxford, 1954, p. 23. MR**0067125 (16:673c)****[6]**R. Ito,*Minimal entropy for endomorphisms of the circle*, preprint, Waseda University, Tokyo, Japan. MR**667298 (84b:58093)****[7]**-,*Rotation sets are closed*, Math. Proc. Cambridge Philos. Soc.**89**(1981), 107-111. MR**591976 (82i:58061)****[8]**S. Newhouse, J. Palis and F. Takens,*Stable families of dynamical systems*I:*Diffeomorphisms*, Inst. Hautes Etudes Sci. Publ. Math. (to appear). MR**699057 (84g:58080)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F20,
28D20,
54C70

Retrieve articles in all journals with MSC: 58F20, 28D20, 54C70

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684649-7

Article copyright:
© Copyright 1983
American Mathematical Society