Devaney's chaos implies existence of -scrambled sets

Author:
Jie-Hua Mai

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2761-2767

MSC (2000):
Primary 54H20; Secondary 37B20, 37D45

DOI:
https://doi.org/10.1090/S0002-9939-04-07514-8

Published electronically:
April 21, 2004

MathSciNet review:
2054803

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a complete metric space without isolated points, and let be a continuous map. In this paper we prove that if is transitive and has a periodic point of period , then has a scrambled set consisting of transitive points such that each is a synchronously proximal Cantor set, and is dense in . Furthermore, if is sensitive (for example, if is chaotic in the sense of Devaney), with being a sensitivity constant, then this is an -scrambled set.

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Additional Information

**Jie-Hua Mai**

Affiliation:
Institute of Mathematics, Shantou University, Shantou, Guangdong, 515063, People’s Republic of China

Email:
jhmai@stu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-04-07514-8

Keywords:
Transitivity,
sensitivity,
synchronously proximal set,
Li-Yorke's chaos,
Devaney's chaos

Received by editor(s):
December 23, 2002

Published electronically:
April 21, 2004

Additional Notes:
The work was supported by the Special Foundation of National Prior Basis Research of China (Grant No. G1999075108).

Communicated by:
Michael Handel

Article copyright:
© Copyright 2004
American Mathematical Society