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

 

A class of nonsofic multidimensional shift spaces


Author: Ronnie Pavlov
Journal: Proc. Amer. Math. Soc. 141 (2013), 987-996
MSC (2010): Primary 37B50; Secondary 37B10, 37A15
Published electronically: July 31, 2012
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Abstract: In one dimension, sofic shifts are fairly well understood and are special examples of shift spaces which must satisfy very restrictive properties. However, in multiple dimensions there are very few known conditions which guarantee nonsoficity of a shift space. In this paper, we show that for any $ \mathbb{Z}^d$ sofic shift $ X$ which satisfies a uniform mixing condition called block gluing in all directions $ \vec {e_2}, \ldots , \vec {e_d}$, the set of legal rows of $ X$ in the $ \vec {e_1}$-direction has a synchronizing word. This allows us to define a (new) large class of nonsofic $ \mathbb{Z}^d$ shift spaces.


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Additional Information

Ronnie Pavlov
Affiliation: Department of Mathematics, University of Denver, 2360 S. Gaylord Street, Denver, Colorado 80208
Email: rpavlov@du.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11382-6
PII: S 0002-9939(2012)11382-6
Keywords: $\mathbb{Z}^{d}$, shift of finite type, sofic, multidimensional
Received by editor(s): March 25, 2011
Received by editor(s) in revised form: August 2, 2011
Published electronically: July 31, 2012
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.