A characterization of chaotic functions with entropy zero via their maximal scrambled sets

Francisco Balibrea; Víctor Jiménez López

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 3, page 293-298
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Balibrea, Francisco, and Jiménez López, Víctor. "A characterization of chaotic functions with entropy zero via their maximal scrambled sets." Mathematica Bohemica 120.3 (1995): 293-298. <http://eudml.org/doc/247783>.

@article{Balibrea1995,
abstract = {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.},
author = {Balibrea, Francisco, Jiménez López, Víctor},
journal = {Mathematica Bohemica},
keywords = {chaos in the sense of Li and Yorke; maximal scrambled sets; topological entropy; chaotic functions; scrambled sets; chaos in the sense of Li and Yorke; maximal scrambled sets; topological entropy},
language = {eng},
number = {3},
pages = {293-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of chaotic functions with entropy zero via their maximal scrambled sets},
url = {http://eudml.org/doc/247783},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Balibrea, Francisco
AU - Jiménez López, Víctor
TI - A characterization of chaotic functions with entropy zero via their maximal scrambled sets
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 3
SP - 293
EP - 298
AB - 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.
LA - eng
KW - chaos in the sense of Li and Yorke; maximal scrambled sets; topological entropy; chaotic functions; scrambled sets; chaos in the sense of Li and Yorke; maximal scrambled sets; topological entropy
UR - http://eudml.org/doc/247783
ER -

References

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  2. V. V. Fedorenko A. N. Šarkovskii, J. Smítal, 10.1090/S0002-9939-1990-1017846-5, Proc. Amer. Math. Soc. 110 (1990), 141-148. (1990) MR1017846DOI10.1090/S0002-9939-1990-1017846-5
  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. (1991) MR1120591
  4. K. Janková, J. Smítal, 10.1017/S0004972700010157, Bull. Austral. Mat. Soc. 34 (1986), 283-292. (1986) MR0854575DOI10.1017/S0004972700010157
  5. V. Jiménez López, Is Li and Yorke's definition a good tool to measure chaos?, PhD Thesis Universidad de Murcia 1992. (1992) 
  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. (1987) MR1085314
  7. T. Y. Li, J. A. Yorke, 10.2307/2318254, Amer. Math. Monthly 82 (1975), 985-992. (1975) Zbl0351.92021MR0385028DOI10.2307/2318254
  8. C. Preston, Iterates of piecewise monotone mappings on an interval, Lecture Notes in Mathematics 1347. Springer, Berlin, 1988. (1988) Zbl0684.58002MR0969131
  9. A. N. Šarkovskii, On cycles and the structure of a continuous map, Ukrain. Mat. Ž. 17 (1965), 104-111. (In Russian.) (1965) MR0186757
  10. J. Smítal, 10.2307/2000468, Trans. Amer. Mat. Soc. 297 (1986), 269-282. (1986) MR0849479DOI10.2307/2000468

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