# American Institute of Mathematical Sciences

January  2022, 42(1): 285-299. doi: 10.3934/dcds.2021116

## Shadowing for families of endomorphisms of generalized group shifts

 Département de mathématiques, Université du Québec à Montréal, Case postale 8888, Succursale Centre-ville, Montréal, QC, H3C 3P8, Canada

Received  April 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $\Sigma \subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $\Gamma$ is a finitely generated monoid consisting of pairwise commuting cellular automata $\Sigma \to \Sigma$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of $\Gamma$ on $\Sigma$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.

Citation: Xuan Kien Phung. Shadowing for families of endomorphisms of generalized group shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 285-299. doi: 10.3934/dcds.2021116
##### References:
 [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Translated from the Russian by S. Feder Proceedings of the Steklov Institute of Mathematics, American Mathematical Society, 90, Providence, R.I., 1967. [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] G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.  doi: 10.1007/BF02559515. [4] F. Blanchard and A. Maass, Dynamical properties of expansive one-sided cellular automata, Israel J. Math., 99 (1997), 149-174.  doi: 10.1007/BF02760680. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second Revised Edition. With A Preface by David Ruelle. Edited by Jean-René Chazottes. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. [6] T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14034-1. [7] T. Ceccherini-Silberstein and M. Coornaert, On surjunctive monoids, Internat. J. Algebra Comput., 25 (2015), 567-606.  doi: 10.1142/S0218196715500113. [8] N. P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.  doi: 10.1090/proc/13654. [9] F. Fiorenzi, Periodic configurations of subshifts on groups, Internat. J. Algebra Comput., 19 (2009), 315-335.  doi: 10.1142/S0218196709005123. [10] B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735.  doi: 10.1017/S0143385700005290. [11] P. Kurka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems, 17 (1997), 417-433.  doi: 10.1017/S014338579706985X. [12] 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. [13] J. v. Neumann, Theory of Self-Reproducing Automata, Univerity of Illinois Press, 1966. [14] P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.  doi: 10.4064/cm110-2-8. [15] A. V. Osipov and S. B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.  doi: 10.1080/14689367.2014.902037. [16] X. K. Phung, On dynamical finiteness properties of algebraic group shifts, to appear in Israel J. Math., arXiv: 2010.04035. [17] S. Y. Pilyugin and S. B. Tikhomirov, Shadowing in actions of some abelian groups, Fund. Math., 179 (2003), 83-96.  doi: 10.4064/fm179-1-7. [18] V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951. [19] A. Schlette, Artinian, almost abelian groups and their groups of automorphisms, Pacific J. Math., 29 (1969), 403-425.  doi: 10.2140/pjm.1969.29.403. [20] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995. [21] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The Structure of Attractors in Dynamical Systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 1978,231–244.

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##### References:
 [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Translated from the Russian by S. Feder Proceedings of the Steklov Institute of Mathematics, American Mathematical Society, 90, Providence, R.I., 1967. [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] G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.  doi: 10.1007/BF02559515. [4] F. Blanchard and A. Maass, Dynamical properties of expansive one-sided cellular automata, Israel J. Math., 99 (1997), 149-174.  doi: 10.1007/BF02760680. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second Revised Edition. With A Preface by David Ruelle. Edited by Jean-René Chazottes. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. [6] T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14034-1. [7] T. Ceccherini-Silberstein and M. Coornaert, On surjunctive monoids, Internat. J. Algebra Comput., 25 (2015), 567-606.  doi: 10.1142/S0218196715500113. [8] N. P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.  doi: 10.1090/proc/13654. [9] F. Fiorenzi, Periodic configurations of subshifts on groups, Internat. J. Algebra Comput., 19 (2009), 315-335.  doi: 10.1142/S0218196709005123. [10] B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735.  doi: 10.1017/S0143385700005290. [11] P. Kurka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems, 17 (1997), 417-433.  doi: 10.1017/S014338579706985X. [12] 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. [13] J. v. Neumann, Theory of Self-Reproducing Automata, Univerity of Illinois Press, 1966. [14] P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.  doi: 10.4064/cm110-2-8. [15] A. V. Osipov and S. B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.  doi: 10.1080/14689367.2014.902037. [16] X. K. Phung, On dynamical finiteness properties of algebraic group shifts, to appear in Israel J. Math., arXiv: 2010.04035. [17] S. Y. Pilyugin and S. B. Tikhomirov, Shadowing in actions of some abelian groups, Fund. Math., 179 (2003), 83-96.  doi: 10.4064/fm179-1-7. [18] V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951. [19] A. Schlette, Artinian, almost abelian groups and their groups of automorphisms, Pacific J. Math., 29 (1969), 403-425.  doi: 10.2140/pjm.1969.29.403. [20] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995. [21] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The Structure of Attractors in Dynamical Systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 1978,231–244.
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