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Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors


Authors: María Isabel Cortez, Fabien Durand and Samuel Petite
Journal: Proc. Amer. Math. Soc. 138 (2010), 1033-1046
MSC (2010): Primary 37B50
DOI: https://doi.org/10.1090/S0002-9939-09-10139-9
Published electronically: November 2, 2009
MathSciNet review: 2566569
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Abstract: In this paper we prove linearly repetitive Delone systems have finitely many Delone system factors up to conjugacy. This result is also applicable to linearly repetitive tiling systems.


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  • [Ad] B. Adamczewski, Codages de rotations et phénomènes d'autosimilarité, J. Th. Nombres de Bordeaux 14 (2002), 351-386. MR 2040682 (2005b:37013)
  • [AD] B. Adamczewski, D. Damanik, Linearly recurrent circle map subshifts and an application to Schrödinger operators, Ann. Henri Poincaré 3 (2002), 1019-1047. MR 1937613 (2004c:37017)
  • [Du1] F. Durand, Linearly recurrent subshifts have a finite number of nonperiodic subshift factors, Ergod. Th. and Dynam. Sys. 20 (2000), 1061-1078. MR 1779393 (2001m:37022)
  • [Du2] F. Durand, Corrigendum and addendum to: Linearly recurrent subshifts have a finite number of nonperiodic subshift factors, Ergod. Th. and Dynam. Sys. 23 (2003), 663-669. MR 1972245 (2004c:37021)
  • [DHS] F. Durand, B. Host, C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergod. Th. and Dynam. Sys. 19 (1999), 953-993. MR 1709427 (2000i:46062)
  • [FKS] H. Furstenberg, H. B. Keynes, L. Shapiro, Prime flows in topological dynamics, Israel J. of Math. 14 (1973), 26-38. MR 0321055 (47:9588)
  • [HRS] C. Holton, C. Radin, L. Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys. 254 (2005), 343-359. MR 2117629 (2006m:37021)
  • [KN] H. B. Keynes, D. Newton, Real prime flows, Trans. Amer. Math. Soc. 217 (1976), 237-255. MR 0400189 (53:4024)
  • [LP] J. C. Lagarias, P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergod. Th. and Dynam. Sys. 23 (2003), 831-867. MR 1992666 (2005a:52018)
  • [Pe] K. Petersen, Factor maps between tiling dynamical systems, Forum Math. 11 (1999), 503-512. MR 1699171 (2000f:37019)
  • [RS] C. Radin, L. Sadun, Isomorphism of hierarchical structures, Ergod. Th. and Dynam. Sys. 21 (2001), 1239-1248. MR 1849608 (2002e:37021)
  • [Ro] E. A. Robinson, Jr., Symbolic dynamics and tilings of $ \mathbb{R}^d$, Symbolic dynamics and its applications, 81-119, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004. MR 2078847 (2005h:37036)
  • [Ru] D. J. Rudolph, Markov tilings of $ \mathbb{R}^n$ and representations of $ \mathbb{R}^n$ actions, Measure and measurable dynamics (Rochester, NY, 1987), 271-290, Contemp. Math., 94, Amer. Math. Soc., Providence, RI, 1989. MR 1012996 (91b:28016)
  • [So1] B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. and Dynam. Sys. 17 (1997), 695-738. MR 1452190 (98f:52030)
  • [So2] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom. 20 (1998), 265-279. MR 1637896 (99f:52028)
  • [So3] B. Solomyak, Spectrum of dynamical systems arising from Delone sets, In: Quasicrystals and discrete geometry, Amer. Math. Soc., Providence, RI, 1998, 265-275. MR 1636783 (99f:58176)

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Additional Information

María Isabel Cortez
Affiliation: Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Avenida Libertador Bernardo O’Higgins 3363, Santiago, Chile
Email: maria.cortez@usach.cl

Fabien Durand
Affiliation: Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex, France
Email: fabien.durand@u-picardie.fr

Samuel Petite
Affiliation: Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS-UMR 6140, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex, France
Email: samuel.petite@u-picardie.fr

DOI: https://doi.org/10.1090/S0002-9939-09-10139-9
Keywords: Delone sets, tiling systems, factor maps, linearly repetitive, Vorono\"{\i } cell.
Received by editor(s): December 8, 2008
Received by editor(s) in revised form: July 29, 2009
Published electronically: November 2, 2009
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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