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For graph maps, one scrambled pair implies Li-Yorke chaos


Authors: Sylvie Ruette and L’ubomír Snoha
Journal: Proc. Amer. Math. Soc. 142 (2014), 2087-2100
MSC (2010): Primary 37E25; Secondary 37B05, 54H20
DOI: https://doi.org/10.1090/S0002-9939-2014-11937-X
Published electronically: March 7, 2014
MathSciNet review: 3182027
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Abstract: For a dynamical system $ (X,f)$, $ X$ being a compact metric space with metric $ d$ and $ f$ being a continuous map $ X\to X$, a set $ S\subseteq X$ is scrambled if every pair $ (x,y)$ of distinct points in $ S$ is scrambled, i.e.,

$\displaystyle \liminf _{n\to +\infty }d(f^n(x),f^n(y))=0 \hbox { and } \limsup _{n\to +\infty }d(f^n(x),f^n(y))>0.$

The system $ (X,f)$ is Li-Yorke chaotic if it has an uncountable scrambled set. It is known that for interval and circle maps, the existence of a scrambled pair implies Li-Yorke chaos, in fact, the existence of a Cantor scrambled set. We prove that the same result holds for graph maps. We further show that on compact countable metric spaces one scrambled pair implies the existence of an infinite scrambled set.

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Additional Information

Sylvie Ruette
Affiliation: Laboratoire de Mathématiques, Bâtiment 425, CNRS UMR 8628, Université Paris-Sud 11, 91405 Orsay cedex, France
Email: Sylvie.Ruette@math.u-psud.fr

L’ubomír Snoha
Affiliation: Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia
Email: Lubomir.Snoha@umb.sk

DOI: https://doi.org/10.1090/S0002-9939-2014-11937-X
Keywords: Scrambled pair, scrambled set, Li-Yorke chaos, graph, countable metric space
Received by editor(s): May 24, 2012
Received by editor(s) in revised form: July 11, 2012
Published electronically: March 7, 2014
Additional Notes: Most of this work was done while the second author was visiting Université Paris-Sud 11 at Orsay. The invitation and the kind hospitality of this institution are gratefully acknowledged. The second author was also partially supported by VEGA grant 1/0978/11 and by the Slovak Research and Development Agency under contract No. APVV-0134-10.
Dedicated: Dedicated to Jaroslav Smítal on the occasion of his 70th birthday
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.