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)



Minimal periodic orbits and topological entropy of interval maps

Author: Bau-Sen Du
Journal: Proc. Amer. Math. Soc. 100 (1987), 482-484
MSC: Primary 58F20; Secondary 26A18, 54C70, 58F08
MathSciNet review: 891150
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any two integers $ m \geq 0$ and $ n \geq 1$, we construct continuous functions from $ \left[ {0,1} \right]$ into itself which have exactly one minimal periodic orbit of least period $ {2^m}\left( {2n + 1} \right)$, but with topological entropy equal to $ \infty $.

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

  • [1] R. Adler, A. Konheim, and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 0175106 (30:5291)
  • [2] L. Alseda, J. Llibre, and R. Serra, Minimal periodic orbits for continuous maps of the interval, Trans. Amer. Math. Soc. 286 (1984), 595-627. MR 760976 (86c:58124)
  • [3] L. Block, J. Guckenheimer, M. Misiurewicz, and L.-S. Young, Periodic points and topological entropy of one dimensional maps, Lecture Notes in Math., vol. 819, Springer-Verlag, Berlin and New York, 1980, pp. 18-34. MR 591173 (82j:58097)
  • [4] W. A. Coppel, Šarkovskii-minimal orbits, Math. Proc. Cambridge Philos. Soc. 93 (1983), 397-408. MR 698345 (84h:58120)
  • [5] C.-W. Ho, On the structure of the minimum orbits of periodic points for maps of the real line (to appear).
  • [6] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. 27 (1979), 167-169. MR 542778 (81b:58033)
  • [7] Z. Nitecki, Maps of the interval with closed periodic set, Proc. Amer. Math. Soc. 85 (1982), 451-456. MR 656122 (83k:58067)
  • [8] -, Topological dynamics on the interval, Progress in Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1-73. MR 670074 (84g:54051)
  • [9] P. Stefan, A theorem of Sharkovsky on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248. MR 0445556 (56:3894)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F20, 26A18, 54C70, 58F08

Retrieve articles in all journals with MSC: 58F20, 26A18, 54C70, 58F08

Additional Information

Keywords: Least period, periodic points, periodic orbits, minimal periodic orbits, topological entropy
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society