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September  2003, 9(5): 1323-1327. doi: 10.3934/dcds.2003.9.1323

Omega-chaos almost everywhere

Jaroslav Smítal 1, and  Marta Štefánková 1,

1. 

Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic, Czech Republic

Received  August 2002 Revised  January 2003 Published  June 2003

Developping ideas of S. Li [Tran. Amer. Math. Soc. 301 (1993), 243--249] concerning the notion of $\omega$-chaos we prove that any transitive continuous map $f$ of the interval is conjugate to a map $g$ of the interval which possesses an $\omega$-scrambled set $S$ of full Lebesgue measure. Thus, for any distinct $x, y$ in $S$, $\omega _g (x)\cap\omega _g(y)$ is non-empty, and $\omega _g(x) \setminus\omega _g(y)$ is uncountable.
Keywords: transitive map., $\omega$-chaos.
Mathematics Subject Classification: Primary: 26A18, 37D45, 37E05; Secondary: 54H20, 26A30, 37A2.
Citation: Jaroslav Smítal, Marta Štefánková. Omega-chaos almost everywhere. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1323-1327. doi: 10.3934/dcds.2003.9.1323
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