Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-17T21:34:06.849Z Has data issue: false hasContentIssue false
  • English
  • Français

There are no chaotic mappings with residual scrambled sets

Published online by Cambridge University Press:  17 April 2009

T. Gedeon
T. Gedeon
Affiliation:
Katedra algebry, Faculty of Mathematics and Physics, Komensky University, 842 15 Bratislava, Czecholovakia.
Rights & Permissions [Opens in a new window]

Extract

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.

A continuous mapping f of a compact real interval I to itself which is chaotic in the sense of Li and Yorke, cannot have a scrambled set residual in an subinterval of I. This shows that chaos in this case cannot be large from a topological point of view.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 3 , December 1987 , pp. 411 - 416
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bruckner, A. M. and Hu, Thakyin, “On scrambled sets for chaotic functions”, (Preprint 1985).Google Scholar
[2]Janková, K. and Smital, J., “Characterization of chaos”, Bull. Auetral. Math. Soc. 34 (1986), 283–292.CrossRefGoogle Scholar
[3]Kan, I., “A chaotic function possessing a scrambled set of positive Lebesgue measure”, Proc. Amer. Math. Soc. 92 (1984), 45–49.CrossRefGoogle Scholar
[4]Li, T. Y. and Yorke, J. A., “Period three implies chaos”, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[5]Misiurewicz, M., “Chaos almost everywhere”, Iteration Theory and its Functional Equations (Ed. Liedl, R. et al. ), Lecture Notes in Math. 1163 (Springer 1985), pp. 125–130.CrossRefGoogle Scholar
[6]Smital, J., “A chaotic function with some extremal properties”, Proc. Amer. Math. Soc. 87 (1983), 54–56.CrossRefGoogle Scholar
[7]Smital, J., “A chaotic function with scrambled set of positive Lebesgue measure”, Proc. Amer. Math. Soc. 92 (1984), 50–54.CrossRefGoogle Scholar
[8]Smital, J., “Chaotic functions with zero topological entropy”, Trans. Amer. Math. Soc. 297 (1986), 269–282.CrossRefGoogle Scholar