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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

Classification of sofic projective subdynamics of multidimensional shifts of finite type


Authors: Ronnie Pavlov and Michael Schraudner
Journal: Trans. Amer. Math. Soc. 367 (2015), 3371-3421
MSC (2010): Primary 37B50; Secondary 37B10, 37B40
DOI: https://doi.org/10.1090/S0002-9947-2014-06259-4
Published electronically: November 4, 2014
MathSciNet review: 3314811
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Abstract: Motivated by Hochman's notion of subdynamics of a $ \mathbb{Z}^d$ subshift (2009), we define and examine the projective subdynamics of $ \mathbb{Z}^d$ shifts of finite type (SFTs) where we restrict not only the action but also the phase space. We show that any $ \mathbb{Z}$ sofic shift of positive entropy is the projective subdynamics of a $ \mathbb{Z}^2$ ( $ \mathbb{Z}^d$) SFT, and that there is a simple condition characterizing the class of zero-entropy $ \mathbb{Z}$ sofic shifts which are not the projective subdynamics of any $ \mathbb{Z}^2$ SFT. We define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory, and discuss how our results fit into this framework. One-dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization, whereas $ \mathbb{Z}$ SFTs only allow for stable realizations and a particular class of zero-entropy proper $ \mathbb{Z}$ sofics only allows for an unstable realization. Finally, we prove that the union of finitely many $ \mathbb{Z}^k$ subshifts, all of which are realizable in $ \mathbb{Z}^d$ SFTs, is again realizable when it contains at least two periodic points, that the projective subdynamics of $ \mathbb{Z}^2$ SFTs with the uniform filling property (UFP) are always stable, thus sofic, and we exhibit a class of non-sofic $ \mathbb{Z}$ subshifts which are not the projective subdynamics of any $ \mathbb{Z}^d$ SFT.


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Additional Information

Ronnie Pavlov
Affiliation: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
Email: rpavlov@du.edu

Michael Schraudner
Affiliation: Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile
Email: mschraudner@dim.uchile.cl

DOI: https://doi.org/10.1090/S0002-9947-2014-06259-4
Keywords: $\mathbb{Z}^d$, multidimensional shift of finite type, sofic systems, subsystem, projective subdynamics, uniform filling property, Sturmian shifts, periodic points
Received by editor(s): January 4, 2012
Received by editor(s) in revised form: May 6, 2013
Published electronically: November 4, 2014
Additional Notes: The second author was partially supported by Basal project CMM, Universidad de Chile, by FONDECYT projects 3080008 and 1100719 and by CONICYT Proyecto Anillo ACT 1103.
Article copyright: © Copyright 2014 by the authors