Hostname: page-component-5c6d5d7d68-txr5j Total loading time: 0 Render date: 2024-08-18T23:29:37.755Z Has data issue: false hasContentIssue false

On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

Published online by Cambridge University Press:  30 September 2009

XAVIER BRESSAUD
Affiliation:
Institut de Mathématiques de Luminy, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: bressaud@iml.univ-mrs.fr)
FABIEN DURAND
Affiliation:
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80000 Amiens, France (email: fabien.durand@u-picardie.fr)
ALEJANDRO MAASS
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universidad de Chile, Avenida Blanco Encalada 2120, Santiago, Chile (email: amaass@dim.uchile.cl)

Abstract

In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bressaud, X., Durand, F. and Maass, A.. Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems. J. Lond. Math. Soc. 72(3) (2005), 799816.CrossRefGoogle Scholar
[2]Cortez, M. I., Durand, F., Host, B. and Maass, A.. Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems. J. Lond. Math. Soc. 67(3) (2003), 790804.Google Scholar
[3]Downarowicz, T. and Lacroix, Y.. A non-regular Toeplitz flow with preset pure point spectrum. Stud. Math. 120(3) (1996), 235246.Google Scholar
[4]Downarowicz, T. and Maass, A.. Finite rank Bratteli–Vershik diagrams are expansive. Ergod. Th. & Dynam. Sys. 28(3) (2008), 739747.CrossRefGoogle Scholar
[5]Downarowicz, T.. Survey of Odometers and Toeplitz Flows (Algebraic and Topological Dynamics, Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005,pp. 737.Google Scholar
[6]Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 10611078.Google Scholar
[7]Gjerde, R. and Johansen, O.. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20 (2000), 16871710.CrossRefGoogle Scholar
[8]Gjerde, R. and Johansen, O.. Bratteli–Vershik models for Cantor minimal systems associated to interval exchange transformations. Math. Scand. 90 (2002), 87100.Google Scholar
[9]Herman, R. H., Putnam, I. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3 (1992), 827864.Google Scholar
[10]Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6 (1986), 529540.Google Scholar