Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-04T06:38:47.512Z Has data issue: false hasContentIssue false
  • English
  • Français

Large chaos in smooth functions of zero topological entropy

Published online by Cambridge University Press:  17 April 2009

Víctor Jiménez López
Víctor Jiménez López
Affiliation:
Departamento de Matemáticas Campus de Espinardo, Universidad de Murcia, 30100 Murcia, Spain
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.

For any α ∈ (0, 1), examples of C∞ functions fα: [0, 1] → [0, 1] with zero topological entropy and possessing a δ−scrambled set of Lebesgue measure α are given. This answers a question posed by Smítal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 2 , October 1992 , pp. 271 - 285
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Alder, R.L., Konheim, A.G. and McAndrew, M.H., ‘Topological entropy’, Trans. Amer. Math. Soc. 114 (1965), 309–319.Google Scholar
[2]Gallego, F. Balibrea and López, V. Jiménez, ‘Weak and strong chaoticity in some classes of functions’, preprint (1990).Google Scholar
[3]Bowen, R. and Franks, J., ‘The periodic points of maps of the disk and the interval’, Topology 15 (1976), 337–342.CrossRefGoogle Scholar
[4]Bruckner, A.M. and Hu, T., ‘On scrambled sets and chaotic functions’, Trans. Amer. Math. Soc. 301 (1987), 289–297.CrossRefGoogle Scholar
[5]Collet, P. and Eckmann, J.P., Iterated maps on the interval as dynamical systems: Progress in physics (Birkhauser, Boston, 1980).Google Scholar
[6]Janková, K. and Smítal, J., ‘A characterization of chaos’, Bull. Austral. Math. Soc. 34 (1986), 283–292.CrossRefGoogle Scholar
[7]Kan, I., ‘A chaotic function possessing a scrambled set of positive Lebesgue measure’, Proc. Amer. Math. Soc. 92 (1984), 45–49.CrossRefGoogle Scholar
[8]Kuchtaand, M.Smítal, J., ‘Two point scrambled set implies chaos’, in European Conference on Iteration Theory (ECIT 87) (World. Sci. Publishing Co., Singapore, 1989).Google Scholar
[9]Li, T.Y. and Yorke, J.A., ‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[10]Martens, M., de Melo, W. and Van Strien, S., ‘Julia-Fatou-Sullivan theory for real one-dimensional dynamical’, Reports of the Faculties of Technical Mathematics and Informatics (Delft, 1988), 88–100.Google Scholar
[11]Misiurewicz, M., ‘Horseshoes for mappings of the interval’, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 167–169.Google Scholar
[12]Misiurewicz, M., ‘Chaos almost everywhere’, Iteration Theory and its Functional Equations, in Lecture notes in Mathematics 1163, pp. 125–130 (Springer-Verlag, Berlin, Heidelberg, New York, 1985).Google Scholar
[13]Misiurewicz, M. and Smítal, J., ‘Smooth chaotic maps with zero topological entropy’, Ergodic Theory Dynamical Systems 8 (1988), 421–424.CrossRefGoogle Scholar
[14]Smítal, J., ‘A chaotic function with a scrambled set of positive Lebesgue measure’, Proc. Amer. Math. Soc. 92 (1984), 50–54.CrossRefGoogle Scholar
[15]Smítal, J., ‘Chaotic functions with zero topological entropy’, Trans. Amer. Math. Soc. 297 (1986), 269–282.CrossRefGoogle Scholar
[16]Šarkovskii, A.N., ‘Coexistence of cycles of a continuous map of the line into itself’, (Russian), Ukrain. Math. Z. 16 (1964), 61–71.Google Scholar
[17]Xiong, J., ‘A chaotic map with zero topological entropy’, Act. Math. Sci. 6, 4 (1986), 439–443.CrossRefGoogle Scholar