Skew-product for group-valued edge labellings of Brateli diagrams. (English) Zbl 1231.37008

Summary: We associate a Cantor dynamical system to a non-properly ordered Bratteli diagram. Group valued edge labellings \(\lambda\) of a Bratteli diagram \(B\) give rise to a skew-product Bratteli diagram \(B(\lambda)\) on which the group acts. The quotient by the group action of the associated dynamics can be a nontrivial extension of the dynamics of \(B\). We exhibit a Bratteli diagram for this quotient and construct a morphism to \(B\) with unique path lifting property. This is shown to be an isomorphism for the dynamics if a property “loops lifting to loops” is satisfied by \(B(\lambda )\to B\).


37B10 Symbolic dynamics
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
Full Text: DOI arXiv Euclid EuDML


[1] J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated \(C^ *\)-algebras, Ergodic Theory Dynam. Systems 18(3) (1998), 509\Ndash537. · Zbl 1053.46520
[2] T. Downarowicz, J. Kwiatkowski, and Y. Lacroix, A criterion for Toeplitz flows to be topologically isomorphic and applications, Colloq. Math. 68(2) (1995), 219\Ndash228. · Zbl 0820.28009
[3] T. Downarowicz and Y. Lacroix, Almost \(1\)-\(1\) extensions of Furstenberg-Weiss type and applications to Toeplitz flows, Studia Math. 130(2) (1998), 149\Ndash170. · Zbl 0916.28013
[4] F. Durand, B. Host, and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19(4) (1999), 953\Ndash993. · Zbl 1044.46543
[5] R. Gjerde and Ø. Johansen, Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows, Ergodic Theory Dynam. Systems 20(6) (2000), 1687\Ndash1710. · Zbl 0992.37008
[6] R. H. Herman, I. F. Putnam, and C. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3(6) (1992), 827\Ndash864. · Zbl 0786.46053
[7] A. Kumjian and D. Pask, \(C^ *\)-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19(6) (1999), 1503\Ndash1519. · Zbl 0949.46034
[8] H. Matui, Finite order automorphisms and dimension groups of Cantor minimal systems, J. Math. Soc. Japan 54(1) (2002), 135\Ndash160. · Zbl 1029.37003
[9] K. Medynets, Cantor aperiodic systems and Bratteli diagrams, C. R. Math. Acad. Sci. Paris 342(1) (2006), 43\Ndash46. · Zbl 1081.37004
[10] S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67(1) (1984), 95\Ndash107. · Zbl 0584.28007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.