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Eigenvalues of Toeplitz minimal systems of finite topological rank

Published online by Cambridge University Press:  05 August 2014

FABIEN DURAND
Affiliation:
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France email fabien.durand@u-picardie.fr
ALEXANDER FRANK
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Avenida Blanco Encalada 2120, Santiago, Chile email afrank@dim.uchile.cl
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 paper we characterize measure-theoretical eigenvalues of Toeplitz Bratteli–Vershik minimal systems of finite topological rank which are not associated to a continuous eigenfunction. Several examples are provided to illustrate the different situations that can occur.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

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References

Bezuglyi, S., Kwiatkowski, J., Medynets, K. and Solomyak, B.. Finite rank Bratteli diagrams: structure of invariant measures. Trans. Amer. Math. Soc. 365 (2013), 26372679.CrossRefGoogle Scholar
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 (2005), 799816.CrossRefGoogle Scholar
Bressaud, X., Durand, F. and Maass, A.. On the eigenvalues of finite rank Bratteli–Vershik dynamical systems. Ergod. Th. & Dynam. Sys. 30 (2010), 639664.CrossRefGoogle Scholar
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 (2003), 790804.CrossRefGoogle Scholar
Dekking, F. M.. The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrsch. Verw. Gebiete 41 (1977/78), 221239.CrossRefGoogle Scholar
Downarowicz, T. and Lacroix, Y.. A non-regular Toeplitz flow with preset pure point spectrum. Studia Math. 120 (1996), 235246.Google Scholar
Downarowicz, T. and Maass, A.. Finite-rank Bratteli–Vershik diagrams are expansive. Ergod. Th. & Dynam. Sys. 28 (2008), 739747.CrossRefGoogle Scholar
Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20 (2000), 10611078.CrossRefGoogle Scholar
Durand, F.. Corrigendum and addendum to ‘Linearly recurrent subshifts have a finite number of non-periodic subshift factors’. Ergod. Th. & Dynam. Sys. 23 (2003), 663669.CrossRefGoogle Scholar
Durand, F.. Combinatorics on Bratteli diagrams and dynamical systems. Combinatorics, Automata and Number Theory (Encyclopedia of Mathematical Applications, 135). Cambridge University Press, Cambridge, 2010, pp. 324372.CrossRefGoogle Scholar
Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19 (1999), 953993.CrossRefGoogle Scholar
Gjerde, R. and Johansen, Ø.. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20 (2000), 16871710.CrossRefGoogle Scholar
Herman, R., Putnam, I. and Skau, C.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3 (1992), 827864.CrossRefGoogle Scholar
Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6 (1986), 529540.CrossRefGoogle Scholar
Iwanik, A.. Toeplitz flows with pure point spectrum. Studia Math. 118 (1996), 2735.CrossRefGoogle Scholar
Jacobs, K. and Keane, M.. 0–1-sequences of Toeplitz type. Z. Wahrsch. Verw. Gebiete 13 (1969), 123131.CrossRefGoogle Scholar
Williams, S.. Toeplitz minimal flows which are not uniquely ergodic. Z. Wahrsch. Verw. Gebiete 67 (1984), 95107.CrossRefGoogle Scholar