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Tower systems for linearly repetitive Delone sets


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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