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Scrambled sets in shift spaces on a countable alphabet


Authors: Brian E. Raines and Tyler Underwood
Journal: Proc. Amer. Math. Soc. 144 (2016), 217-224
MSC (2010): Primary 37B10, 37B20, 37D40, 54H20
DOI: https://doi.org/10.1090/proc/12690
Published electronically: June 24, 2015
MathSciNet review: 3415590
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Abstract: In this paper we characterize the shift spaces which have Li-Yorke chaos (an uncountable scrambled set). We focus primarily on shifts over a countably infinite alphabet. We represent them as either edge-shifts on an infinite graph (the subshift of finite type case) or as labelled edge-shifts on an infinite graph (the sofic shift case). We show in the setting of a subshift of finite type on a shift over a countable alphabet that the shift space has Li-Yorke chaos if, and only if, it has a single scrambled pair, and in this case the scrambled set is closed and perfect (but not necessarily compact). We give an example of a sofic shift over an infinite alphabet which has a single scrambled pair but does not have Li-Yorke chaos.


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Additional Information

Brian E. Raines
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798–7328
Email: brian_raines@baylor.edu

Tyler Underwood
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798–7328
Address at time of publication: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
Email: tyler_underwood@umail.ucsb.edu

DOI: https://doi.org/10.1090/proc/12690
Keywords: Uncountable chaotic sets, subshifts, transition graph, chaotic pair, Baire space, Li-Yorke chaos, edge-shift, sofic shift, shift of finite type
Received by editor(s): July 7, 2014
Received by editor(s) in revised form: December 2, 2014
Published electronically: June 24, 2015
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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