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Tower systems for linearly repetitive Delone sets

  • JOSÉ ALISTE-PRIETO (a1) and DANIEL CORONEL (a2)
Abstract
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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[CFS82] I. P. Cornfeld , S. V. Fomin and Ya. G. Sinaĭ . Ergodic Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245). Springer, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ; MR 832433(87f:28019).

[Moo97] R. V. Moody (ed.) . The Mathematics of Long-range Aperiodic Order (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 489). Kluwer Academic Publishers Group, Dordrecht, 1997; MR 1460016(98a:52001).

[Sen81] E. Seneta . Nonnegative Matrices and Markov Chains, 2nd edn(Springer Series in Statistics). Springer, New York, 1981; MR 719544(85i:60058).

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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