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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Specification property and distributional chaos almost everywhere
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by Piotr Oprocha and Marta Štefánková PDF
Proc. Amer. Math. Soc. 136 (2008), 3931-3940 Request permission

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
  • MR Author ID: 765606
  • ORCID: 0000-0002-0261-7229
  • Email: oprocha@agh.edu.pl
  • Marta Štefánková
  • Affiliation: Mathematical Institute, Silesian University, 74601 Opava, Czech Republic
  • Email: marta.stefankova@math.slu.cz
  • Received by editor(s): September 27, 2007
  • Published electronically: June 24, 2008
  • Communicated by: Jane M. Hawkins
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 3931-3940
  • MSC (2000): Primary 37B05; Secondary 54H20
  • DOI: https://doi.org/10.1090/S0002-9939-08-09602-0
  • MathSciNet review: 2425733