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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Rotation topological factors of minimal $ \mathbb{Z}^{d}$-actions on the Cantor set


Authors: Maria Isabel Cortez, Jean-Marc Gambaudo and Alejandro Maass
Journal: Trans. Amer. Math. Soc. 359 (2007), 2305-2315
MSC (2000): Primary 54H20; Secondary 52C23
Published electronically: December 20, 2006
MathSciNet review: 2276621
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Abstract: In this paper we study conditions under which a free minimal $ \mathbb{Z}^d$-action on the Cantor set is a topological extension of the action of $ d$ rotations, either on the product $ \mathbb{T}^d$ of $ d$ $ 1$-tori or on a single $ 1$-torus $ \mathbb{T}^1$. We extend the notion of linearly recurrent systems defined for $ \mathbb{Z}$-actions on the Cantor set to $ \mathbb{Z}^d$-actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.


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Additional Information

Maria Isabel Cortez
Affiliation: Departamento de Ingeniería Matemática, Fac. Ciencias Físicas y Matemáticas, Universidad de Chile, Av. Blanco Encalada 2120 5to piso, Santiago, Chile – and – Institut de Mathématiques de Bourgogne, U.M.R. CNRS 5584, Université de Bourgogne, U.F.R. des Sciences et Téchniques, B.P. 47870- 21078 Dijon Cedex, France
Address at time of publication: Departamento de Matemática, Universidad de Santiago de Chile, Avenida Alameda Libertador O’Higgins 3363, Codigo Postal 7254758, Santiago, Chile

Jean-Marc Gambaudo
Affiliation: Centro de Modelamiento Matemático, U.M.R. CNRS 2071, Av. Blanco Encalada 2120, 7to piso, Santiago, Chile
Address at time of publication: Université de Nice - Sophia Antipolis, Laboratoire J.-A. Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice cedex 2, France

Alejandro Maass
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Fac. Ciencias Físicas y Matemáticas, Universidad de Chile, Av. Blanco Encalada 2120 5to piso, Santiago, Chile
Email: amaass@dim.uchile.cl

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04027-X
PII: S 0002-9947(06)04027-X
Keywords: Minimal Cantor free actions, linearly recurrent systems, rotation factors
Received by editor(s): August 25, 2004
Received by editor(s) in revised form: March 24, 2005
Published electronically: December 20, 2006
Article copyright: © Copyright 2006 American Mathematical Society