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Tower systems for linearly repetitive Delone sets

Published online by Cambridge University Press:  23 November 2010

JOSÉ ALISTE-PRIETO  and
DANIEL CORONEL
JOSÉ ALISTE-PRIETO
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120, 7to. piso. Santiago, Chile (email: jaliste@dim.uchile.cl)
DANIEL CORONEL
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile (email: acoronel@mat.puc.cl)
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Abstract

In this paper we study linearly repetitive Delone sets and prove, following the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly repetitive Delone set admits a properly nested sequence of box decompositions (tower system) with strictly positive and uniformly bounded (in size and norm) transition matrices. This generalizes a result of Durand for linearly recurrent symbolic systems. Furthermore, we apply this result to give a new proof of a classic estimation of Lagarias and Pleasants on the rate of convergence of patch frequencies.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1595 - 1618
Copyright
Copyright © Cambridge University Press 2010

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