Real Analysis Exchange

Scrambled Sets for Transitive Maps

Marek Lampart

Full-text: Open access


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

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

First available in Project Euclid: 2 June 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


Lampart, Marek. Scrambled Sets for Transitive Maps. Real Anal. Exchange 27 (2001), no. 2, 801--808.

Export citation


  • M. Babilonová, The bitransitive continuous maps of the interval are conjugate to maps extremely chaotic a.e., Acta Math. Univ. Comenianae LXIX, 2 (2000), 229–232.
  • M. Babilonová, Extreme chaos and transitivity, Preprint MA 21/2000, Mathematical Institute, Silesian University, Opava. Internat. J. Bifur. Chaos Appl. Sci. Engrg. –- to appear.
  • L. S. Block and W. A. Coppel, Dynamics in one dimension,
  • A. M. Blokh, The "Spectral" Decomposition for one–dimensional Maps, Dynam. Reported, 4 (1995), 1–59.
  • A. M. Bruckner and T. Hu, On Scrambled sets for chaotic functions, Trans. Amer. Math. Soc., 301 (1987), 289–297.
  • W. J. Gorman, The homeomorphic lTransformation of C–sets Into D–sets, Proc. Amer. Math. Soc., 17 (1966), 825–830.
  • I. Kan, A chaotic functions possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc., 92 (1984), 45–49.
  • S. Li, $\omega$–chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243–249.
  • M. Misiurewicz, Chaos almost everywhere, Iteration Theory and its Functional Equations, Lecture Notes in Math., 1163, Springer, Berlin 1985, 125–130.
  • J. Smítal, A chaotic functions with some extremal properties, Proc. Amer. Math. Soc., 87 1983), 54–56.
  • J. Smítal, A chaotic functions with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc., 92 (1984), 50–54.