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On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

Published online by Cambridge University Press:  30 September 2009

XAVIER BRESSAUD ,
FABIEN DURAND  and
ALEJANDRO MAASS
XAVIER BRESSAUD
Affiliation:
Institut de Mathématiques de Luminy, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected])
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: [email protected])
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: [email protected])
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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
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 639 - 664
Copyright
Copyright © Cambridge University Press 2009

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