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Devaney's chaos implies existence of $s$-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
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Abstract: Let $X$ be a complete metric space without isolated points, and let $f:X\to X$ be a continuous map. In this paper we prove that if $f$ is transitive and has a periodic point of period $p$, then $f$ has a scrambled set $S=\bigcup _{n=1}^{\infty }C_{n}$consisting of transitive points such that each $C_{n}$ is a synchronously proximal Cantor set, and $\bigcup _{i=0}^{p-1}f^{i}(S)$ is dense in $X$. Furthermore, if $f$ is sensitive (for example, if $f$ is chaotic in the sense of Devaney), with $2s$ being a sensitivity constant, then this $S$ is an $s$-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

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