Matrix characterization of multidimensional subshifts of finite type

[EN] Let X ⊂ AZd be a 2-dimensional subshift of finite type. We prove that any 2-dimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general d-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space X to exhibit periodic points.The first author thanks National Board for Higher Mathematics (NBHM) Grant No. 2/48(39)/2016/NBHM(R.P)/R&D II/4519 for financial support.Sharma, P.; Kumar, D. (2019). Matrix characterization of multidimensional subshifts of finite type. Applied General Topology. 20(2):407-418. https://doi.org/10.4995/agt.2019.11541407418202J. C. Ban, W. G. Hu, S. S. Lin and Y. H. Lin, Verification of mixing properties in two-dimensional shifts of finite type, arXiv:1112.2471v2.M.-P. Beal, F. Fiorenzi and F. Mignosi, Minimal forbidden patterns of multi-dimensional shifts, Int. J. Algebra Comput. 15 (2005), 73-93. https://doi.org/10.1142/S0218196705002165R. Berger, The undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966). https://doi.org/10.1090/memo/0066M. Boyle, R. Pavlov and M. Schraudner, Multidimensional sofic shifts without separation and their factors, Transactions of the American Mathematical Society 362, no. 9 (2010), 4617-4653. https://doi.org/10.1090/S0002-9947-10-05003-8X.-C. Fu, W. Lu, P. Ashwin and J. Duan, Symbolic representations of iterated maps, Topological Methods in Nonlinear Analysis 18 (2001), 119-147. https://doi.org/10.12775/TMNA.2001.027J. Hadamard, Les surfaces a coubures opposees et leurs lignes geodesiques, J. Math. Pures Appi. 5 IV (1898), 27-74.M. Hochman, On dynamics and recursive properties of multidimensional symbolic dynamics, Invent. Math. 176:131 (2009). https://doi.org/10.1007/s00222-008-0161-7M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics 171, no. 3 (2010), 2011-2038. https://doi.org/10.4007/annals.2010.171.2011B. P. Kitchens, Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-3-642-58822-8_7S. Lightwood, Morphisms from non-periodic $Z^2$-subshifts I: Constructing embeddings from homomorphisms, Ergodic Theory Dynam. Systems 23, no. 2 (2003), 587-609. https://doi.org/10.1017/S014338570200130XD. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511626302A. Quas and P. Trow, Subshifts of multidimensional shifts of finite type, Ergodic Theory and Dynamical Systems 20, no. 3 (2000), 859-874. https://doi.org/10.1017/S0143385700000468R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209. https://doi.org/10.1007/BF01418780C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948), 379-423, 623-656. https://doi.org/10.1002/j.1538-7305.1948.tb00917.xH. H. Wicke and J. M. Worrell, Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J. 34 (1967), 255-271. https://doi.org/10.1215/S0012-7094-67-03430-