Tokyo Journal of Mathematics

Chaos and Entropy for Circle Maps

Megumi MIYAZAWA

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Abstract

Our aim is to check that the notions of positive entropy, chaos in the sense of Devaney and $\omega$-chaos are equivalent for the circle maps.

Article information

Source
Tokyo J. Math., Volume 25, Number 2 (2002), 453-458.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208864

Digital Object Identifier
doi:10.3836/tjm/1244208864

Mathematical Reviews number (MathSciNet)
MR1948675

Zentralblatt MATH identifier
1028.37027

Subjects
Primary: 37E10: Maps of the circle
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Citation

MIYAZAWA, Megumi. Chaos and Entropy for Circle Maps. Tokyo J. Math. 25 (2002), no. 2, 453--458. doi:10.3836/tjm/1244208864. https://projecteuclid.org/euclid.tjm/1244208864


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References

  • L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics 5 (1993), World Scientific.
  • J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332–334.
  • L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics 1513 (1992), Springer.
  • L. Block, E. Coven, I. Mulvey and Z. Nitecki, Homoclinic and non-wandering points for maps of the circle, Ergod. Th. and Dynam. Sys., 3 (1983), 521–532.
  • R. L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings (1986).
  • E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067–1075.
  • R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107–111.
  • M. Kuchta, Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolin., 31 (1990), 383–390.
  • S. Li, Dynamical properties of the shift maps on the inverse limit spaces, Ergod. Th. and Dynam. Sys., 12 (1992), 95–108.
  • S. Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243–249.
  • T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985–992.
  • M. Misiurewicz, Twist sets for maps of the circle, Ergod. Th. and Dynam. Sys., 4 (1984), 391–404.
  • S. Silverman, On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math., 22 (1992), 353–375.
  • J. Smí tal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269–282.
  • M. Vellekoop and R. Berglund, On intervals, transitivity $ = $ chaos, Amer. Math. Monthly, 101 (1994), 353–355.
  • J. Xiong, The attracting centre of a continuous self-map of the interval, Ergod. Th. and Dynam. Sys., 8, (1988), 205–213.
  • J. Xiong and Z. Yang, Chaos caused by a topologically mixing map, Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, 9 World Sci. (1991), 550–572.