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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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

Author(s): María Isabel Cortez; Fabien Durand; Samuel Petite
Journal: Proc. Amer. Math. Soc. 138 (2010), 1033-1046.
MSC (2010): Primary 37B50
Posted: November 2, 2009
MathSciNet review: 2566569
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Abstract | References | Similar articles | Additional information

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.

References:

[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: 10.1090/S0002-9939-09-10139-9
PII: S 0002-9939(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
Posted: November 2, 2009
Communicated by: Bryna Kra
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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