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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Distributional chaos revisited


Author: Piotr Oprocha
Journal: Trans. Amer. Math. Soc. 361 (2009), 4901-4925
MSC (2000): Primary 37B99; Secondary 37D45, 37B10, 37B20
Published electronically: April 13, 2009
MathSciNet review: 2506431
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Abstract: In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case.

In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in $ \mathbb{R}^3$. Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics.

We also prove that any interval map with positive topological entropy contains two invariant subsets $ X,Y \subset I$ such that $ f\vert _X$ has positive topological entropy and $ f\vert _Y$ displays distributional chaos of type $ 1$, but not conversely.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04810-7
PII: S 0002-9947(09)04810-7
Keywords: Distributional chaos, chaotic pair, Li-Yorke chaos
Received by editor(s): October 17, 2007
Published electronically: April 13, 2009
Dedicated: Dedicated to Professor Jaroslav Smítal on the occasion of his 65th birthday.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.