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)



The chain recurrent set for maps of the interval

Authors: Louis Block and John E. Franke
Journal: Proc. Amer. Math. Soc. 87 (1983), 723-727
MSC: Primary 58F22; Secondary 54H20
MathSciNet review: 687650
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a continuous map of a compact interval into itself. We show that if the set of periodic points of $ f$ is a closed set then every chain recurrent point is periodic.

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

  • [1] L. Block, Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67 (1977), 357-360. MR 0467841 (57:7692)
  • [2] -, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978). 576-580. MR 509258 (81m:58063)
  • [3] C. Conley, The gradient structure of a flow. I, IBM Research, RC 3932 (#17806), Yorktown Heights, N. Y., July 17, 1972.
  • [4] -, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1976.
  • [5] E. M. Coven and G. A. Hedlund, Continuous maps of the interval whose periodic points form a closed set, Proc. Amer. Math. Soc. 79 (1980), 127-133. MR 560598 (81a:54042)
  • [6] E. M. Coven and Z. Nitecki, Nonwandering sets of the powers of maps of the interval, Ergodic Theory Dynamical Systems 1 (1981), 9-31. MR 627784 (82m:58043)
  • [7] T. Li, M. Misiurewicz, G. Pianigiani and J. Yorke, Odd chaos, preprint. MR 643455 (83d:58058)
  • [8] Z. Nitecki, Maps of the interval with closed periodic set, Proc. Amer. Math. Soc. 85 (1982), 451-456. MR 656122 (83k:58067)
  • [9] -, Topological dynamics on the interval, Ergodic Theory and Dynamical Systems II (College Park. Md. 1979-1980), Progress in Math., vol. 21, Birkhauser, Boston, 1982, pp. 1-73. MR 670074 (84g:54051)
  • [10] Jin-Cheng Ziong, Continuous self-maps of the closed interval whose periodic points form a closed set, J. China Univ. Sci. and Tech. 11 (1981), 14-23. MR 701781 (84h:58124a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F22, 54H20

Retrieve articles in all journals with MSC: 58F22, 54H20

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society