American Institute of Mathematical Sciences

February  2021, 41(2): 1005-1021. doi: 10.3934/dcds.2020307

Recoding Lie algebraic subshifts

 Department of Mathematics and Statistics, University of Turku, Turku, Finland

* Corresponding author: Ilkka Törmä

Ville Salo supported by Academy of Finland grant 2608073211.

Received  December 2019 Revised  June 2020 Published  February 2021 Early access  August 2020

Fund Project: Ilkka Törmä supported by Academy of Finland grant 295095

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.

Citation: Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
References:
 [1] S. Barbieri, R. Gómez, B. 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.

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References:
 [1] S. Barbieri, R. Gómez, B. 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.
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