| 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 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
120 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
293-298 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 |
| DOI:
|
10.21136/MB.1995.126005 |
| . |
| Date available:
|
2009-09-24T21:11:56Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126005 |
| . |
| 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, 10.1080/00029890.1975.11994008 |
| 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, 10.1007/BFb0079771 |
| 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 |
| . |
