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

 

Positive topological entropy implies chaos DC2


Author: T. Downarowicz
Journal: Proc. Amer. Math. Soc. 142 (2014), 137-149
MSC (2010): Primary 37A35
Published electronically: August 28, 2013
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Abstract: Using methods of entropy in ergodic theory, we prove that positive topological entropy implies chaos DC2. That is, if a system $ (X,T)$ has positive topological entropy, then there exists an uncountable set $ E$ such that for any two distinct points $ x,y$ in $ E$,

$\displaystyle \liminf _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)=0 \ \ \ $$\displaystyle \text {and} \ \ \limsup _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)>0 . $


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Additional Information

T. Downarowicz
Affiliation: Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: downar@pwr.wroc.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11717-X
PII: S 0002-9939(2013)11717-X
Keywords: Distributional chaos, chaos DC2, ergodic process, scrambled set
Received by editor(s): October 8, 2011
Received by editor(s) in revised form: February 20, 2012
Published electronically: August 28, 2013
Additional Notes: The author’s research was supported from resources for science in years 2009-2012 as research project grant MENII N N201 394537, Poland
Communicated by: Byrna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.