Positive topological entropy implies chaos DC2
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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 $E$, \[ \liminf _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)=0 \ \ \ \text {and} \ \ \limsup _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)>0 . \]References
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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
- MR Author ID: 59525
- Email: downar@pwr.wroc.pl
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 137-149
- MSC (2010): Primary 37A35
- DOI: https://doi.org/10.1090/S0002-9939-2013-11717-X
- MathSciNet review: 3119189