Scrambled Sets for Transitive Maps

Abstract

We deal with two types of chaos: the well known chaos in the sense of Li and Yorke and $\omega$-chaos which was introduced by S. Li in 1993. In this paper we prove that every bitransitive map $f \in C(I,I)$ is conjugate to $g \in C(I,I)$, which satisfies the following conditions,

1. there is a $c$-dense $\omega$-scrambled set for $g$,

2. there is an extremely LY-scrambled set for $g$ with full Lebesgue measure,

3. every $\omega$-scrambled set of $g$ has zero Lebesgue measure.