Real Analysis Exchange

Scrambled Sets for Transitive Maps

Marek Lampart

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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.

Article information

Source
Real Anal. Exchange, Volume 27, Number 2 (2001), 801-808.

Dates
First available in Project Euclid: 2 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1212412877

Mathematical Reviews number (MathSciNet)
MR1923170

Zentralblatt MATH identifier
1062.37026

Subjects
Primary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 37D45: Strange attractors, chaotic dynamics 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 26A30: Singular functions, Cantor functions, functions with other special properties

Keywords
LY-chaos $\omega$-chaos scrambled sets transitive maps

Citation

Lampart, Marek. Scrambled Sets for Transitive Maps. Real Anal. Exchange 27 (2001), no. 2, 801--808. https://projecteuclid.org/euclid.rae/1212412877


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