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Journal of the London Mathematical Society Advance Access originally published online on March 12, 2008
Journal of the London Mathematical Society 2008 78(1):1-20; doi:10.1112/jlms/jdn002
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© 2008 London Mathematical Society

G-odometers and their almost one-to-one extensions

María Isabel Cortez

Departamento de Matemática y Ciencia de
la Computación
Universidad de Santiago de Chile
Av. Libertador Bernardo O’Higgins 3363
Santiago
Chile
mcortez@usach.cl

Samuel Petite

Laboratoire Amiénois de Mathématique
Fondamentale et Appliquée
CNRS UMR 6140
Université de Picardie Jules Verne
33 rue Saint-Leu
8039 Amiens cedex 1
France

In this paper we recall the concepts of G-odometers and G-subodometers for G-actions, where G is a discrete finitely generated group; these generalize the notion of an odometer in the case G = Z. We characterize the G-regularly recurrent systems as the minimal almost one-to-one extensions of subodometers, from which we deduce that the family of the G-Toeplitz subshifts coincides with the family of the minimal symbolic almost one-to-one extensions of subodometers. We determine the continuous eigenvalues of these systems. When G is amenable and residually finite, a characterization of the G-invariant measures of these systems is given.

2000 Mathematics Subject Classification 54H20 (primary), 37B50 (secondary).

Received January 29, 2007; revised November 22, 2007; published online March 12, 2008.


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