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 & 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.  Google Scholar

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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

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G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.  doi: 10.1007/BF02559515.  Google Scholar

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F. Blanchard and A. Maass, Dynamical properties of expansive one-sided cellular automata, Israel J. Math., 99 (1997), 149-174.  doi: 10.1007/BF02760680.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T. Ceccherini-Silberstein and M. Coornaert, On surjunctive monoids, Internat. J. Algebra Comput., 25 (2015), 567-606.  doi: 10.1142/S0218196715500113.  Google Scholar

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F. Fiorenzi, Periodic configurations of subshifts on groups, Internat. J. Algebra Comput., 19 (2009), 315-335.  doi: 10.1142/S0218196709005123.  Google Scholar

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B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735.  doi: 10.1017/S0143385700005290.  Google Scholar

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P. Kurka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems, 17 (1997), 417-433.  doi: 10.1017/S014338579706985X.  Google Scholar

[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.  Google Scholar

[13] J. v. Neumann, Theory of Self-Reproducing Automata, Univerity of Illinois Press, 1966.   Google Scholar
[14]

P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.  doi: 10.4064/cm110-2-8.  Google Scholar

[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.  Google Scholar

[16]

X. K. Phung, On dynamical finiteness properties of algebraic group shifts, to appear in Israel J. Math., arXiv: 2010.04035. Google Scholar

[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.  Google Scholar

[18]

V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951. Google Scholar

[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.  Google Scholar

[20]

K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[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.  Google Scholar

show all references

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.  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]

G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.  doi: 10.1007/BF02559515.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

T. Ceccherini-Silberstein and M. Coornaert, On surjunctive monoids, Internat. J. Algebra Comput., 25 (2015), 567-606.  doi: 10.1142/S0218196715500113.  Google Scholar

[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.  Google Scholar

[9]

F. Fiorenzi, Periodic configurations of subshifts on groups, Internat. J. Algebra Comput., 19 (2009), 315-335.  doi: 10.1142/S0218196709005123.  Google Scholar

[10]

B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735.  doi: 10.1017/S0143385700005290.  Google Scholar

[11]

P. Kurka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems, 17 (1997), 417-433.  doi: 10.1017/S014338579706985X.  Google Scholar

[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.  Google Scholar

[13] J. v. Neumann, Theory of Self-Reproducing Automata, Univerity of Illinois Press, 1966.   Google Scholar
[14]

P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460.  doi: 10.4064/cm110-2-8.  Google Scholar

[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.  Google Scholar

[16]

X. K. Phung, On dynamical finiteness properties of algebraic group shifts, to appear in Israel J. Math., arXiv: 2010.04035. Google Scholar

[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.  Google Scholar

[18]

V. Salo, When are group shifts of finite type?, preprint, arXiv: 1807.01951. Google Scholar

[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.  Google Scholar

[20]

K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[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.  Google Scholar

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