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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Classification of sofic projective subdynamics of multidimensional shifts of finite type
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by Ronnie Pavlov and Michael Schraudner PDF
Trans. Amer. Math. Soc. 367 (2015), 3371-3421

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
  • MR Author ID: 845553
  • 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
  • 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.
  • © Copyright 2014 by the authors
  • 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
  • MathSciNet review: 3314811