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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Specification property and distributional chaos almost everywhere


Authors: Piotr Oprocha and Marta Stefánková
Journal: Proc. Amer. Math. Soc. 136 (2008), 3931-3940
MSC (2000): Primary 37B05; Secondary 54H20
Published electronically: June 24, 2008
MathSciNet review: 2425733
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Abstract: Our main result shows that a continuous map $ f$ acting on a compact metric space $ (X,\rho )$ with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set $ S$ dense in $ X$ which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points $ u,v\in S$, the upper distribution function is identically 1 and the lower distribution function is zero at some $ \varepsilon >0$. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when $ X$ is the $ k$-dimensional cube $ I^{k}$. If $ X=I$, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.


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Additional Information

Piotr Oprocha
Affiliation: Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: oprocha@agh.edu.pl

Marta Stefánková
Affiliation: Mathematical Institute, Silesian University, 74601 Opava, Czech Republic
Email: marta.stefankova@math.slu.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09602-0
PII: S 0002-9939(08)09602-0
Received by editor(s): September 27, 2007
Published electronically: June 24, 2008
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.