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Title: A characterization of chaotic functions with entropy zero via their maximal scrambled sets (English)
Author: Balibrea, Francisco
Author: Jiménez López, Víctor
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 120
Issue: 3
Year: 1995
Pages: 293-298
Summary lang: English
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Category: math
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Summary: In this note we characterize chaotic functions (in the sense of Li and Yorke) with topological entropy zero in terms of the structure of their maximal scrambled sets. In the interim a description of all maximal scrambled sets of these functions is also found. (English)
Keyword: chaos in the sense of Li and Yorke
Keyword: maximal scrambled sets
Keyword: topological entropy
Keyword: chaotic functions
Keyword: scrambled sets
MSC: 26A18
MSC: 37D45
MSC: 54H20
MSC: 58F13
idZBL: Zbl 0852.54039
idMR: MR1369687
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Date available: 2009-09-24T21:11:56Z
Last updated: 2015-09-04
Stable URL: http://hdl.handle.net/10338.dmlcz/126005
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Reference: [1] R. L. Alder A. G. Konheim, M. H. McAndrew: Topological entropy.Trans. Amer. Mat. Soc. 114 (1965), 309-319. MR 0175106, 10.1090/S0002-9947-1965-0175106-9
Reference: [2] V. V. Fedorenko A. N. Šarkovskii, J. Smítal: Characterizations of weakly chaotic maps of the interval.Proc. Amer. Math. Soc. 110 (1990), 141-148. MR 1017846, 10.1090/S0002-9939-1990-1017846-5
Reference: [3] V. V. Fedorenko, J. Smítal: Maps of the interval Ljapunov stable on the set of nonwandering points.Acta Math. Univ. Comenian. (N. S.) 60 (1991), 11-14. MR 1120591
Reference: [4] K. Janková, J. Smítal: A characterization of chaos.Bull. Austral. Mat. Soc. 34 (1986), 283-292. MR 0854575, 10.1017/S0004972700010157
Reference: [5] V. Jiménez López: Is Li and Yorke's definition a good tool to measure chaos?.PhD Thesis Universidad de Murcia 1992.
Reference: [6] M. Kuchta, J. Smítal: Two point scrambled set implies chaos.Proceedings of the European Conference on Iteration Theory (ECIT 87), Caldes de Malavella, Spain, 1987. World Sci. Publishing, Singapore, 1989, pp. 427-430. MR 1085314
Reference: [7] T. Y. Li, J. A. Yorke: Period three implies chaos.Amer. Math. Monthly 82 (1975), 985-992. Zbl 0351.92021, MR 0385028, 10.2307/2318254
Reference: [8] C. Preston: Iterates of piecewise monotone mappings on an interval.Lecture Notes in Mathematics 1347. Springer, Berlin, 1988. Zbl 0684.58002, MR 0969131
Reference: [9] A. N. Šarkovskii: On cycles and the structure of a continuous map.Ukrain. Mat. Ž. 17 (1965), 104-111. (In Russian.) MR 0186757
Reference: [10] J. Smítal: Chaotic functions with zero topological entropy.Trans. Amer. Mat. Soc. 297 (1986), 269-282. MR 0849479, 10.2307/2000468
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