Simplicity of 2-graph algebras associated to Dynamical Systems

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $\Lambda$ we consider has an associated $C^*$-algebra, denoted $C^*(\Lambda)$, which is simple and purely infinite when $\Lambda$ is aperiodic. We give new, straightforward conditions which ensure that $\Lambda$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $\Lambda$ in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.Comment: 19 page

Minimal Absent Words in Rooted and Unrooted Trees

We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet of cardinality. We show that the set of minimal absent words of a rooted (resp. unrooted) tree T with n nodes has cardinality (resp.), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time (resp. assuming an integer alphabet of size polynomial in n

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.2019.11541SWORD407418202J. 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-

Minimal forbidden patterns of multi-dimensional shifts

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