## 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.## References

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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