Garden of Eden and weakly periodic points for certain expansive actions of groups. (English) Zbl 1521.37007
Summary: We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri et al. [Adv. Math. 397, Article ID 108196, 52 p. (2022; Zbl 1491.37010)], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of H. Li [Ergodic Theory Dyn. Syst. 39, No. 11, 3075–3088 (2019; Zbl 1421.37005)]
(where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of D. B. Cohen [Adv. Math. 308, 599–626 (2017; Zbl 1400.20034)] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37B40 | Topological entropy |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
Keywords:
expansive dynamical systems; Garden of Eden theorem; entropy; periodic points; amenable groupsReferences:
| [1] | Barbieri, S., García-Ramos, F. and Li, H.. Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math.397 (2022), 52. · Zbl 1491.37010 |
| [2] | Barbieri, S., Gómez, R., Marcus, B. and Taati, S.. Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups. Nonlinearity33 (2020), 2409-2454. · Zbl 1487.37040 |
| [3] | Bartholdi, L.. Gardens of Eden and amenability on cellular automata. J. Eur. Math. Soc. (JEMS)12 (2010), 241-248. · Zbl 1185.37020 |
| [4] | Bartholdi, L.. Amenability of groups is characterized by Myhill’s theorem. J. Eur. Math. Soc. (JEMS)21 (2019), 3191-3197. With an appendix by D. Kielak. · Zbl 1458.37017 |
| [5] | Berger, R.. The undecidability of the domino problem. Mem. Amer. Math. Soc.66 (1966), 72. · Zbl 0199.30802 |
| [6] | Capobianco, S., Kari, J. and Taati, S.. An ‘almost dual’ to Gottschalk’s conjecture. Cellular Automata and Discrete Complex Systems(Lecture Notes in Computer Science, 9664). Eds. M. Cook and T. Neary. Springer, Cham, 2016, pp. 77-89. · Zbl 1369.68261 |
| [7] | Ceccherini-Silberstein, T. and Coornaert, M.. The Garden of Eden theorem for linear cellular automata. Ergod. Th. & Dynam. Sys.26 (2006), 53-68. · Zbl 1085.37008 |
| [8] | Ceccherini-Silberstein, T. and Coornaert, M.. Cellular Automata and Groups(Springer Monographs in Mathematics). Springer-Verlag, Berlin, 2010. · Zbl 1218.37004 |
| [9] | Ceccherini-Silberstein, T. and Coornaert, M.. Expansive actions on uniform spaces and surjunctive maps. Bull. Math. Sci.1 (2011), 79-98. · Zbl 1256.54054 |
| [10] | Ceccherini-Silberstein, T. and Coornaert, M.. The Myhill property for strongly irreducible subshifts over amenable groups. Monatsh. Math.165 (2012), 155-172. · Zbl 1283.37022 |
| [11] | Ceccherini-Silberstein, T. and Coornaert, M.. A garden of Eden theorem for Anosov diffeomorphisms on tori. Topology Appl.212 (2016), 49-56. · Zbl 1366.37071 |
| [12] | Ceccherini-Silberstein, T. and Coornaert, M.. Expansive actions with specification on uniform spaces, topological entropy, and the Myhill property. J. Dyn. Control Syst.27 (2021), 427-456. · Zbl 1480.37019 |
| [13] | Ceccherini-Silberstein, T., Coornaert, M. and Li, H.. Homoclinically expansive actions and a Garden of Eden theorem for harmonic models. Comm. Math. Phys.368 (2019), pp. 1175-1200. · Zbl 1423.37013 |
| [14] | Ceccherini-Silberstein, T., Coornaert, M. and Li, H.. Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity. Preprint, 2021, arXiv:2107.12047. |
| [15] | Ceccherini-Silberstein, T., Coornaert, M. and Phung, X. K.. On the Garden of Eden theorem for endomorphisms of symbolic algebraic varieties. Pacific J. Math.306 (2020), 31-66. · Zbl 1468.37013 |
| [16] | Ceccherini-Silberstein, T. G., Machì, A. and Scarabotti, F.. Amenable groups and cellular automata. Ann. Inst. Fourier (Grenoble)49 (1999), 673-685. · Zbl 0920.43001 |
| [17] | Chandgotia, N., Han, G., Marcus, B., Meyerovitch, T. and Pavlov, R.. One-dimensional Markov random fields, Markov chains and topological Markov fields. Proc. Amer. Math. Soc.142 (2014), 227-242. · Zbl 1283.37017 |
| [18] | Chung, N.-P. and Lee, K.. Topological stability and pseudo-orbit tracing property of group actions. Proc. Amer. Math. Soc.146 (2018), 1047-1057. · Zbl 1384.37032 |
| [19] | Chung, N.-P. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math.199 (2015), 805-858. · Zbl 1320.37009 |
| [20] | Cohen, D. B.. The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math.308 (2017), 599-626. · Zbl 1400.20034 |
| [21] | Cohen, D. B. and Goodman-Strauss, C.. Strongly aperiodic subshifts on surface groups. Groups Geom. Dyn.11 (2017), 1041-1059. · Zbl 1377.37028 |
| [22] | Culik, K. Ii and Kari, J.. An aperiodic set of Wang cubes. STACS 96 (Grenoble, 1996)(Lecture Notes in Computer Science, 1046). Eds. C. Puech and R. Reischuk. Springer, Berlin, 1996, pp. 137-146. · Zbl 1379.68120 |
| [23] | Doucha, M. and Gismatullin, J.. On Dual surjunctivity and applications. Preprint, 2020, arXiv:2008.10565. Groups Geom. Dyn., to appear. |
| [24] | Druţu, C. and Kapovich, M.. Geometric Group Theory(American Mathematical Society Colloquium Publications, 63). American Mathematical Society, Providence, RI, 2018. With an appendix by B. Nica. · Zbl 1447.20001 |
| [25] | Engelking, R.. General Topology(Sigma Series in Pure Mathematics, 6), 2nd edn. Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. · Zbl 0684.54001 |
| [26] | Fiorenzi, F.. The Garden of Eden theorem for sofic shifts. Pure Math. Appl. (PU.M.A.)11 (2000), 471-484. · Zbl 0980.37004 |
| [27] | Fiorenzi, F.. Cellular automata and strongly irreducible shifts of finite type. Theoret. Comput. Sci.299 (2003), 477-493. · Zbl 1042.68077 |
| [28] | Gottschalk, W.. Some general dynamical notions. Recent Advances in Topological Dynamics (Proceedings of the Conference on Topological Dynamics, Held at Yale University 1972, in Honor of Gustav Arnold Hedlund)(Lecture Notes in Mathematics, 318). Ed. A. Beck. Springer-Verlag, Berlin, 1973, pp. 120-125. · Zbl 0255.54035 |
| [29] | Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS)1 (1999), 109-197. · Zbl 0998.14001 |
| [30] | Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory3 (1969), 320-375. · Zbl 0182.56901 |
| [31] | Jeandel, E.. Aperiodic subshifts of finite type on groups. Preprint, 2015, arXiv:1501.06831. |
| [32] | Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems(Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020 |
| [33] | Kerr, D. and Li, H.. Ergodic Theory: Independence and Dichotomies(Springer Monographs in Mathematics). Springer, Cham, 2016. · Zbl 1396.37001 |
| [34] | Li, H.. Garden of Eden and specification. Ergod. Th. & Dynam. Sys.39 (2019), 3075-3088. · Zbl 1421.37005 |
| [35] | Meyerovitch, T.. Pseudo-orbit tracing and algebraic actions of countable amenable groups. Ergod. Th. & Dynam. Sys.39 (2019), 2570-2591. · Zbl 1431.37030 |
| [36] | Moore, E. F.. Machine models of self-reproduction. Mathematical Problems in the Biological Sciences(Proceedings of Symposia in Applied Mathematics, 14). American Mathematical Society, Providence, RI, 1963, pp. 17-34. · Zbl 0126.32408 |
| [37] | Mozes, S.. Aperiodic tilings. Invent. Math.128 (1997), 603-611. · Zbl 0879.52011 |
| [38] | Myhill, J.. The converse of Moore’s Garden-of-Eden theorem. Proc. Amer. Math. Soc.14 (1963), 685-686. · Zbl 0126.32501 |
| [39] | Nasu, M.. Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property. Trans. Amer. Math. Soc.352 (2000), 4731-4757. · Zbl 0954.37013 |
| [40] | Osipov, A. V. and Tikhomirov, S. B.. Shadowing for actions of some finitely generated groups. Dyn. Syst.29 (2014), 337-351. · Zbl 1350.37030 |
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