Lebesgue Measure of Recurrent Scrambled Sets

Abstract

It was proved by M. Babilonová-Štefánková (Int J Bifurc Chaos 13(7):1695–1700, 2003) that each bitransitive continuous map f of the interval is conjugated to a map g which is distributionally chaotic with a distributionally scrambled set D. The goal of this chapter is to improve this result, by showing that D is formed by points that are recurrent but not almost periodic. Moreover, as a main result it will be proved that any bitransitive map $f \in C(I,I)$ is topologically conjugate to a map $g \in C(I,I)$ which satisfies the following conditions: (i) g is extremally Li–Yorke chaotic with Li–Yorke scrambled set S with full Lebesgue measure and $S \subset R(g) \backslash \mathrm{A}(g)$, (ii) g is ω chaotic and every ω scrambled set Ω has zero Lebesgue measure and $\Omega \subset R(g) \backslash \mathrm{A}(g)$, (iii) g is distributionally chaotic with a distributionally scrambled set D with full Lebesgue measure and $D \subset R(g) \backslash \mathrm{A}(g)$.