Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-23T06:29:16.499Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous chaotic functions of an interval have generically small scrambled sets

Published online by Cambridge University Press:  17 April 2009

Ivan Mizera
Ivan Mizera
Affiliation:
Department of Mathematics, Komensk University842 15 Bratislava, Czechoslovakia
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.

It is shown that continuous self-mappings of a compact interval, chaotic in the sense of Li and Yorke, have generically, in the uniform topology, only scrambled sets which are nowhere dense and of zero Lebesgue measure.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 89 - 92
Copyright
Copyright Australian Mathematical Society 1988

References

1Block, L., Stability of periodic orbits in the theorem of arkovskiῐ, Proc. Amer. Math. Soc. 81 (1981), 333336.Google Scholar
2Butler, G.J., Pianigiani, G., Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc. 18 (1978), 255265.CrossRefGoogle Scholar
3Bruckner, A.M., Thakyin, Hu, On scrambled sets for chaotic functions, Trans. Amer. Math. Soc. 301 (1987), 289297.CrossRefGoogle Scholar
4Gedeon, T., There are no residual scrambled sets, Bull. Austral. Math. Soc. (to appear).Google Scholar
5Jankov, K., On the stability of chaotic functions, preprint 1984.Google Scholar
6Jankov, K., Smtal, J., A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283292.CrossRefGoogle Scholar
7Kan, I., A chaotic function possessing a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 4549.CrossRefGoogle Scholar
8Kloeden, P.E., Chaotic difference equations are dense, Bull. Austral. Math. Soc. 15 (1976), 371379.CrossRefGoogle Scholar
9Li, T.Y., Yorke, J.A., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985992.CrossRefGoogle Scholar
10Misiurewicz, M., Chaos almost everywhere, in Iteration Theory and its Functional Equations: Lecture Notes in Mathematics 1163, ed. Liedl, R.P., Reich, L.P., Targonski, G. (Springer, Berlin, 1985).Google Scholar
11Misiurewicz, M., Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sr. Sci. Math. 27 (1979), 167169.Google Scholar
12Smtal, J., A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), 5456.CrossRefGoogle Scholar
13Smtal, J., A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 5054.CrossRefGoogle Scholar
14Smtal, J., Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269282.CrossRefGoogle Scholar