# 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  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 & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
##### References:
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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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1981.  Google Scholar [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.  Google Scholar [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.   Google Scholar [8] P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc., 3 (1954), 419-436.  doi: 10.1112/plms/s3-4.1.419.  Google Scholar [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.  Google Scholar [10] D. Kerr and H. Li, Ergodic Theory, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar [11] B. P. Kitchens, Expansive dynamics on zero-dimensional groups, Ergodic Theory Dyn. Syst., 7 (1987), 249-261.  doi: 10.1017/S0143385700003989.  Google Scholar [12] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York, 1998.  Google Scholar [13] B. Malman, Zero-divisors and Idempotents in Group Rings, Master's thesis, Lund University, 2014. Google Scholar [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.  Google Scholar [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.  Google Scholar [16] V. Salo, Subshifts with Simple Cellular Automata, PhD thesis, University of Turku, 2014. Google Scholar [17] V. Salo, Universal groups of cellular automata, preprint, arXiv: 1808.08697. Google Scholar [18] V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951. Google Scholar [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.  Google Scholar [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.  Google Scholar [21] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-0277-2.  Google Scholar
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