Advanced Search
Article Contents
Article Contents

Recoding Lie algebraic subshifts

  • * Corresponding author: Ilkka Törmä

    * Corresponding author: Ilkka Törmä

Ville Salo supported by Academy of Finland grant 2608073211.

Ilkka Törmä supported by Academy of Finland grant 295095

Abstract Full Text(HTML) Related Papers Cited by
  • We study internal Lie algebras in the category of subshifts on a fixed group – or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely many compatible Lie brackets even up to automorphisms of the underlying vector shift, and we state the classification problem of such brackets.

    From attempts to generalize these results to other acting groups, the following questions arise: Does every f.g. group admit a linear cellular automaton of infinite order? Which groups admit abelian group shifts whose homoclinic group is not generated by finitely many orbits? For the first question, we show that the Grigorchuk group admits such a CA, and for the second we show that the lamplighter group admits such group shifts.

    Mathematics Subject Classification: Primary:37B10;Secondary:17B99, 54H15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. BarbieriR. GómezB. Marcus and S. Taati, Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups, Nonlinearity, 33 (2020), 2409-2454.  doi: 10.1088/1361-6544/ab6a75.
    [2] S. Barbieri, F. García-Ramos and H. Li, Markovian properties of continuous group actions: Algebraic actions, entropy and the homoclinic group, preprint, arXiv: 1911.00785.
    [3] F. Blanchard and Y. Lacroix, Zero entropy factors of topological flows, Proc. Amer. Math. Soc., 119 (1993), 985-992.  doi: 10.1090/S0002-9939-1993-1155593-2.
    [4] M. Boyle and M. Schraudner, $\Bbb Z^d$ group shifts and {B}ernoulli factors, Ergodic Theory Dynam. Systems, 28 (2008), 367-387.  doi: 10.1017/S0143385707000697.
    [5] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1981.
    [6] N.-P. Chung and H. Li, Homoclinic groups, IE groups, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858.  doi: 10.1007/s00222-014-0524-1.
    [7] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985. 
    [8] P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc., 3 (1954), 419-436.  doi: 10.1112/plms/s3-4.1.419.
    [9] D. Kerr and H. Li, Combinatorial independence and sofic entropy, Commun. Math. Stat., 1 (2013), 213-257.  doi: 10.1007/s40304-013-0001-y.
    [10] D. Kerr and H. Li, Ergodic Theory, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49847-8.
    [11] B. P. Kitchens, Expansive dynamics on zero-dimensional groups, Ergodic Theory Dyn. Syst., 7 (1987), 249-261.  doi: 10.1017/S0143385700003989.
    [12] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York, 1998.
    [13] B. Malman, Zero-divisors and Idempotents in Group Rings, Master's thesis, Lund University, 2014.
    [14] N. Matte Bon, Topological full groups of minimal subshifts with subgroups of intermediate growth, J. Mod. Dyn., 9 (2015), 67-80.  doi: 10.3934/jmd.2015.9.67.
    [15] T. Meyerovitch, Pseudo-orbit tracing and algebraic actions of countable amenable groups, Ergodic Theory Dynam. Systems, 39 (2019), 2570-2591.  doi: 10.1017/etds.2017.126.
    [16] V. Salo, Subshifts with Simple Cellular Automata, PhD thesis, University of Turku, 2014.
    [17] V. Salo, Universal groups of cellular automata, preprint, arXiv: 1808.08697.
    [18] V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951.
    [19] V. Salo and I. Törmä, On shift spaces with algebraic structure, in How the World Computes, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7318 (2012), 636–645. doi: 10.1007/978-3-642-30870-3_64.
    [20] V. Salo and I. Törmä, Category Theory of Symbolic Dynamics, Theoret. Comput. Sci., 567 (2015), 21-45.  doi: 10.1016/j.tcs.2014.10.023.
    [21] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-0277-2.
  • 加载中

Article Metrics

HTML views(426) PDF downloads(208) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint