Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T19:29:40.198Z Has data issue: false hasContentIssue false
  • English
  • Français

Are chaotic functions really chaotic

Published online by Cambridge University Press:  17 April 2009

Bau-Sen Du
Bau-Sen Du
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, Minnesota 55455, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a class C of continuous functions from [0, 1] onto itself which are chaotic in the sense of Li and Yorke, but with the property that almost all (in the sense of Lebesgue) points of [0, 1] are eventually fixed. For some continuous functions from [0, 1] onto itself which are not in C, We also show that their non-wandering sets are all equal to the interval [0, 1].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 1 , August 1983 , pp. 53 - 66
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Beyer, W.A. and Stein, P.R., “Period doubling for trapezoid function iteration: Metric theory”, Adv. Appl. Math. 3 (1982), 1–17.CrossRefGoogle Scholar
[2]Beyer, W.A. and Stein, P.R., “Further results on periods and period-doubling for iterates of the trapezoid function”, Adv. Appl. Math. 3 (1982), 265–287.CrossRefGoogle Scholar
[3]Butler, G.J. and Pianigiani, G., “Periodic points and chaotic functions in the unit interval”, Bull. Austral. Math. Soc. 18 (1978), 255–265.CrossRefGoogle Scholar
[4]Collet, Pierre, Eckmann, Jean-Pierre, Iterated maps on the interval as dynamical systems (Progress in Physics, 1. Birkhäuser, Basel, Boston, Stuttgart, 1980).Google Scholar
[5]Coppel, W.A., “The solution of equations by iteration”, Proc. Cambridge Philos. Soc. 51 (1955), 41–43.CrossRefGoogle Scholar
[6]Coven, Ethan M. and Hedlund, G.A., “P = R for maps of the interval”, Proc. Amer. Math. Soc. 79 (1980), 316–318.Google Scholar
[7]Du, Bau-Sen, “A chaotic function whose non-wandering set is the Cantor ternary set” (Preprint, University of Minnesota, Minneapolis, 1983).CrossRefGoogle Scholar
[8]Fuglister, Frederick J., “A note on chaos”, J. Combin. Theory Ser. A 26 (1979), 186–188.CrossRefGoogle Scholar
[9]Kloeden, Peter E., “Chaotic difference equations are dense”, Bull. Austral. Math. Soc. 15 (1976), 371–379.CrossRefGoogle Scholar
[10]Li, Tien-Yien and Yorke, James A., “Period three implies chaos”, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[11]Nathanson, Melvyn B., “Permutations, periodicity, and chaos”, J. Combin. Theory Ser. A 22 (1977), 61–68.CrossRefGoogle Scholar
[12]Smítal, J., “A chaotic function with some extremal properties”, Proc. Amer. Math. Soc. 87 (1983), 54–56.CrossRefGoogle Scholar