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Finite-rank Bratteli–Vershik diagrams are expansive

Published online by Cambridge University Press:  01 June 2008

TOMASZ DOWNAROWICZ  and
ALEJANDRO MAASS
TOMASZ DOWNAROWICZ
Affiliation:
Institute of Mathematics, Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: [email protected])
ALEJANDRO MAASS
Affiliation:
Department of Mathematical Engineering and Center of Mathematical Modeling, University of Chile, Av. Blanco Encalada 2120, 5to piso, Santiago, Chile (email: [email protected])
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Abstract

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 739 - 747
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Auslander, J.. Minimal Flows and Their Extensions (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.Google Scholar
[2]Cortez, M., Durand, F., Host, B. and Maass, A.. Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems. J. London Math. Soc. 67 (2003), 790–804.CrossRefGoogle Scholar
[3]Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 1061–1078.CrossRefGoogle Scholar
[4]Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19 (1999), 953–993.Google Scholar
[5]Giordano, T., Putnam, I. and Skau, C.. Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[6]Gjerde, R. and Johansen, O.. Bratteli–Vershik models for Cantor minimal systems: Applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20 (2000), 1687–1710.CrossRefGoogle Scholar
[7]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.Google Scholar
[8]Herman, R., Putnam, I. and Skau, C.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3 (1992), 827–864.Google Scholar