Scrambled sets in shift spaces on a countable alphabet

Raines, Brian; Underwood, Tyler

doi:10.1090/proc/12690

Scrambled sets in shift spaces on a countable alphabet
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by Brian E. Raines and Tyler Underwood

Proc. Amer. Math. Soc. 144 (2016), 217-224

DOI: https://doi.org/10.1090/proc/12690

Published electronically: June 24, 2015

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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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