Matrix characterization of multidimensional subshifts of finite type

• Sharma, Puneet ^[1] ; Kumar, Dileep ^[1]
  1. [1] Indian Institute of Technology Jodhpur
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 20, Nº. 2, 2019, págs. 407-418
• Idioma: inglés
• DOI: 10.4995/agt.2019.11541
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • J. 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....
  • R. Berger, The undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966). https://doi.org/10.1090/memo/0066
  • M. Boyle, R. Pavlov and M. Schraudner, Multidimensional sofic shifts without separation and their factors, Transactions of the American Mathematical...
  • X.-C. Fu, W. Lu, P. Ashwin and J. Duan, Symbolic representations of iterated maps, Topological Methods in Nonlinear Analysis 18 (2001), 119-147....
  • J. 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/s00222-008-0161-7
  • M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Annals of Mathematics 171, no....
  • B. 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_7
  • S. Lightwood, Morphisms from non-periodic $Z^2$-subshifts I: Constructing embeddings from homomorphisms, Ergodic Theory Dynam. Systems 23,...
  • D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511626302
  • A. Quas and P. Trow, Subshifts of multidimensional shifts of finite type, Ergodic Theory and Dynamical Systems 20, no. 3 (2000), 859-874....
  • R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209. https://doi.org/10.1007/BF01418780
  • C. 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.x
  • H. H. Wicke and J. M. Worrell, Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J. 34 (1967), 255-271....