On the periodic points of functions on a manifold

Author:
Chung-wu Ho

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2625-2630

MSC (2000):
Primary 37C25; Secondary 54H25, 58C30

Published electronically:
February 12, 2002

MathSciNet review:
1900870

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Abstract | References | Similar Articles | Additional Information

Abstract: In a conference on fixed point theory, B. Halpern of Indiana University considered the problem of reducing the number of periodic points of a map by homotopy. He also asked whether the number of periodic points of a function could be increased by a homotopy. In this paper, we will show that for any map on a closed manifold, an arbitrarily small perturbation can always create infinitely many periodic points of arbitrarily high periods.

**1.**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****2.**B. Halpern,*The minimal number of periodic points*, Abstr. Amer. Math. Soc., 1 (1980), 775-G8, p. 269.**3.**Chung-Wu Ho,*On Block’s condition for simple periodic orbits of functions on an interval*, Trans. Amer. Math. Soc.**281**(1984), no. 2, 827–832. MR**722777**, 10.1090/S0002-9947-1984-0722777-3**4.**Chung Wu Ho and Charles Morris,*A graph-theoretic proof of Sharkovsky’s theorem on the periodic points of continuous functions*, Pacific J. Math.**96**(1981), no. 2, 361–370. MR**637977****5.**Morris W. Hirsch,*Differential topology*, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR**0448362****6.**Tien Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**0385028****7.**James R. Munkres,*Elementary differential topology*, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR**0198479****8.**Zbigniew Nitecki,*Topological dynamics on the interval*, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR**670074**

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Additional Information

**Chung-wu Ho**

Affiliation:
Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026 – and – Department of Mathematics, Evergreen Valley College, San Jose, California 95135

Email:
cho@siue.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06361-X

Keywords:
Manifolds,
periodic points,
homotopy,
digraphs

Received by editor(s):
February 9, 1999

Received by editor(s) in revised form:
April 1, 2001

Published electronically:
February 12, 2002

Communicated by:
Alan Dow

Article copyright:
© Copyright 2002
American Mathematical Society