Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-02-06361-X
Published electronically: February 12, 2002
MathSciNet review: 1900870
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. L. Block, J. Guckenheimer, M. Misiurewicz, and L. Young, Periodic points and topological entropy of one dimensional maps, Lect. Notes Math., 819 (1980), Springer-Verlag, 18-34. MR 82j:58097
  • 2. B. Halpern, The minimal number of periodic points, Abstr. Amer. Math. Soc., 1 (1980), 775-G8, p. 269.
  • 3. C.W. Ho, On Block's condition for simple periodic orbits of functions on an interval, Trans. Amer. Math. Soc., 281 (1984), 827-832. MR 85g:54035
  • 4. C.W. Ho and C. Morris, A graph-theoretical proof of Sharkovsky's theorem on the periodic points of continuous functions, Pacific J. Math., 96 (1981), 361-370. MR 83d:58056
  • 5. M. Hirsch, Differential Topology, Grad. Texts in Math., Vol. 33, Springer-Verlag, 1976. MR 56:6669
  • 6. T. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. MR 52:5898
  • 7. J. Munkres, Elementary Differential Topology, Revised Ed., Annals of Math. Studies, 54, Princeton Univ. Press, 1966. MR 33:6637
  • 8. Z. Nitecki, Topological dynamics on the interval, Ergodic Theory and Dynamical System II, Progress in Math., Vol. 21, Ed. by A. Katok, Birkhäuser (1982), 1-73. MR 84g:54051

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37C25, 54H25, 58C30

Retrieve articles in all journals with MSC (2000): 37C25, 54H25, 58C30


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

American Mathematical Society