Devaney's chaos implies existence of s-scrambled sets

Proceedings of the American Mathematical Society

Volumn 132, Issue 9, 2004, Pages 2761-2767

  Mai, Jie Hua ^a  

^a SHANTOU UNIVERSITY   (China)

Author keywords

Devaney's chaos; Li Yorke's chaos; Sensitivity; Synchronously proximal set; Transitivity

Indexed keywords

EID: 4344700167     PISSN: 00029939     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9939-04-07514-8     Document Type: Article

References (14)

1. E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic? in: Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., Vol. 5, de Gruyter, Berlin, 1996, pp. 25-40. MR 971:58106
2. J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334. MR 93d:54059
3. N. C. Bernardes, Jr. On the set of points with a dense orbit, Proc. Amer. Math. Soc., 128 (2000), 3421-3423. MR 2001b:54048
4. F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Theory and Dynamical Systems, 20 (2000), 641-662. MR 2002b:37019
5. R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading, MA, 1989. MR 91a:58114
6. X.-C. Fu, Y. Fu, J. Duan and R. S. Mackay, Chaotic properties of subshifts generated by a nonperiodic recurrent orbit, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1067-1073. MR 2001b:37011
7. E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. MR 94j:58109
8. W. Huang and X. Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory and Dynamical Systems, 21 (2001), 77-91. MR 2002d:37020
9. W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology and its Applications, 117 (2002), 259-272. MR 2003b:37017
10. B. Kolev and M.-C. Pérouème, Recurrent surface homeomorphisms, Math. Proc. Cambridge Philos. Soc., 124(1998), 161-168. MR 99j:58162
11. T. Li and J. Yorke, Period 3 implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. MR 52:5898
12. J. H. Mai, Continuous maps with the whole space being a scrambled set, Chinese Sci. Bull., 42 (1997), 1494-1497. MR 99f:54054
13. J. H. Mai, Scrambled sets of continuous maps of 1-dimensional polyhedra, Trans. Amer. Math. Soc., 351 (1999), 353-362. MR 99c:58111
14. 2 are periodic, Proc. Amer. Math. Soc., 110 (1990), 1083-1088. MR 91k:54070