×

Direct topological factorization for topological flows. (English) Zbl 1375.37051

Summary: This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for \(\mathbb{Z}\)-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over \(\mathbb{Z}^{d}\) and other groups, and \(\mathbb{Z}\)-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full \(n\)-shift, the multidimensional \(3\)-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive \(\mathbb{G}\)-action must be finite, but an example is provided of a non-expansive \(\mathbb{Z}\)-action for which there is no finite direct-prime factorization. The question about existence of direct-prime factorization of expansive actions remains open, even for \(\mathbb{G}=\mathbb{Z}\).

MSC:

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F.Blanchard. Fully positive topological entropy and topological mixing. Symbolic Dynamics and its Applications (New Haven, CT, 1991)(Contemporary Mathematics, 135). American Mathematical Society, Providence, RI, 1992, pp. 95-105.10.1090/conm/135/1185082 · Zbl 0783.54033
[2] F.Blanchard and G.Hansel. Systèmes codés. Theoret. Comput. Sci.44(1) (1986), 17-49.10.1016/0304-3975(86)90108-8 · Zbl 0601.68056
[3] R.Bowen. Topological entropy and axiom A. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV, Berkeley, CA, 1968). American Mathematical Society, Providence, RI, 1970, pp. 23-41. · Zbl 0207.54402
[4] M.Boyle, B.Marcus and P.Trow. Resolving maps and the dimension group for shifts of finite type. Mem. Amer. Math. Soc.70(377) (1987), vi+146. · Zbl 0651.54018
[5] M.Boyle, R.Pavlov and M.Schraudner. Multidimensional sofic shifts without separation and their factors. Trans. Amer. Math. Soc.362(9) (2010), 4617-4653.10.1090/S0002-9947-10-05003-8 · Zbl 1207.37011
[6] N.Chandgotia and T.Meyerovitch. Markov random fields, Markov cocycles and the 3-colored chessboard. Preprint, 2013, arXiv:1305.0808. · Zbl 1356.37030
[7] N.-P.Chung and H.Li. Homoclinic groups, i.e. groups, and expansive algebraic actions. Invent. Math.199(3) (2015), 805-858.10.1007/s00222-014-0524-1 · Zbl 1320.37009
[8] H.Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1-49.10.1007/BF01692494 · Zbl 0146.28502 · doi:10.1007/BF01692494
[9] E.Glasner. Ergodic Theory via Joinings(Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.10.1090/surv/101 · Zbl 1038.37002 · doi:10.1090/surv/101
[10] T.Hamachi, K.Inoue and W.Krieger. Subsystems of finite type and semigroup invariants of subshifts. J. Reine Angew. Math.632 (2009), 37-61. · Zbl 1177.37019
[11] M.Hochman and T.Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Ann. of Math. (2)171(3) (2010), 2011-2038.10.4007/annals.2010.171.2011 · Zbl 1192.37022
[12] J.Kari. Representation of reversible cellular automata with block permutations. Math. Syst. Theory29(1) (1996), 47-61.10.1007/BF01201813 · Zbl 0840.68081 · doi:10.1007/BF01201813
[13] W.Krieger. On the uniqueness of the equilibrium state. Math. Syst. Theory8(2) (1974/75), 97-104.10.1007/BF01762180 · Zbl 0302.28011 · doi:10.1007/BF01762180
[14] W.Krieger. On a syntactically defined invariant of symbolic dynamics. Ergod. Th. & Dynam. Sys.20(2) (2000), 501-516.10.1017/S0143385700000249 · Zbl 0992.37010 · doi:10.1017/S0143385700000249
[15] W.Krieger and K.Matsumoto. Zeta functions and topological entropy of the Markov-Dyck shifts. Münster J. Math.4 (2011), 171-183. · Zbl 1258.37017
[16] D. A.Lind. Entropies and factorizations of topological Markov shifts. Bull. Amer. Math. Soc. (N.S.)9(2) (1983), 219-222.10.1090/S0273-0979-1983-15162-5 · Zbl 0524.58034 · doi:10.1090/S0273-0979-1983-15162-5
[17] D. A.Lind. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys.4(2) (1984), 283-300.10.1017/S0143385700002443 · Zbl 0546.58035 · doi:10.1017/S0143385700002443
[18] T.Meyerovitch. Tail invariant measures of the Dyck shift. Israel J. Math.163 (2008), 61-83.10.1007/s11856-008-0004-7 · Zbl 1155.28011 · doi:10.1007/s11856-008-0004-7
[19] K.Schmidt. Automorphisms of compact abelian groups and affine varieties. Proc. Lond. Math. Soc. (3)61(3) (1990), 480-496.10.1112/plms/s3-61.3.480 · Zbl 0789.28013 · doi:10.1112/plms/s3-61.3.480
[20] K.Schmidt. The cohomology of higher-dimensional shifts of finite type. Pacific J. Math.170(1) (1995), 237-269.10.2140/pjm.1995.170.237 · Zbl 0866.28016 · doi:10.2140/pjm.1995.170.237
[21] K.Schmidt. Tilings, fundamental cocycles and fundamental groups of symbolic Z^d-actions. Ergod. Th. & Dynam. Sys.18(6) (1998), 1473-1525.10.1017/S0143385798118060 · Zbl 0915.58030 · doi:10.1017/S0143385798118060
[22] B.Weiss. Actions of amenable groups. Topics in Dynamics and Ergodic Theory(London Mathematical Society Lecture Note Series, 310). Cambridge University Press, Cambridge, 2003, pp. 226-262.10.1017/CBO9780511546716.012 · Zbl 1079.37002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.