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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: [email protected])
DANIEL CORONEL
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile (email: [email protected])
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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.

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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References

[AP98]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C *-algebras. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509–537; MR 1631708(2000a:46112).CrossRefGoogle Scholar
[BBG06]Bellissard, J., Benedetti, R. and Gambaudo, J. M.. Spaces of tilings, finite telescopic approximations and gap-labeling. Comm. Math. Phys. 261(1) (2006), 1–41; MR 2193205(2007c:46063).CrossRefGoogle Scholar
[BDM05]Bressaud, X., Durand, F. and Maass, A.. Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems. J. Lond. Math. Soc. (2) 72(3) (2005), 799–816; MR 2190338(2006j:37011).Google Scholar
[BDM10]Bressaud, X., Durand, F. and Maass, A.. Continuous and measurable eigenvalues of finite rank Bratteli–Vershik dynamical systems. Ergod. Th. & Dynam. Sys. 30(3) (2010), 639–664.CrossRefGoogle Scholar
[Bes08a]Besbes, A.. Uniform ergodic theorems on aperiodic linearly repetitive tilings and applications. Rev. Math. Phys. 20(5) (2008), 597–623; MR 2422207.CrossRefGoogle Scholar
[Bes08b]Besbes, A.. Contributions a l’étude de quelques systèmes quasi-crystallographics (in French). PhD Thesis, Université Pierre et Marie Curie, Paris, 2008.Google Scholar
[BG03]Benedetti, R. and Gambaudo, J. M.. On the dynamics of 𝔾-solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23(3) (2003), 673–691; MR 1992658(2004f:37019).Google Scholar
[CDHM03]Cortez, M. I., Durand, F., Host, B. and Maass, A.. Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems. J. Lond. Math. Soc. (2) 67(3) (2003), 790–804; MR 1967706(2004b:37018).CrossRefGoogle Scholar
[CFS82]Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G.. 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).CrossRefGoogle Scholar
[CGM07]Cortez, M. I., Gambaudo, J. M. and Maass, A.. Rotation topological factors of minimal ℤd-actions of the Cantor set. Trans. Amer. Math. Soc. 359(5) (2007), 2305–2315; MR 2276621(2007k:37010).Google Scholar
[Cor]Coronel, D.. The cohomological equation over dynamical systems arising from Delone sets. Ergod. Th. & Dynam. Sys., doi:10.1017/S0143385710000209.CrossRefGoogle Scholar
[DL06]Damanik, D. and Lenz, D.. Substitution dynamical systems: characterization of linear repetitivity and applications. J. Math. Anal. Appl. 321(2) (2006), 766–780; MR 2241154(2007d:37008).CrossRefGoogle Scholar
[Dur00]Durand, F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20(4) (2000), 1061–1078; MR 1779393(2001m:37022).Google Scholar
[For00]Forrest, A.. A Bratteli diagram for commuting homeomorphisms of the Cantor set. Internat. J. Math. 11(2) (2000), 177–200; MR 1754619(2001d:37008).CrossRefGoogle Scholar
[Ghy99]Ghys, É.. Laminations par surfaces de Riemann. Dynamique et Géométrie Complexes (Lyon, 1997) (Panoramas et Synthèses, 8). Société Mathématique de France, Paris, 1999, pp. 49; 95MR 1760843(2001g:37068).Google Scholar
[GM06]Gambaudo, J.-M. and Martens, M.. Algebraic topology for minimal Cantor sets. Ann. Henri Poincaré 7(3) (2006), 423–446; MR 2226743(2006m:37007).CrossRefGoogle Scholar
[GMPS10]Giordano, T., Matui, H., Putnam, I. F. and Skau, C. F.. Orbit equivalence for Cantor minimal ℤd-systems. Invent. Math. 179(1) (2010), 119–158.Google Scholar
[HPS92]Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(6) (1992), 827–864; MR 1194074(94f:46096).CrossRefGoogle Scholar
[KP00]Kellendonk, J. and Putnam, I. F.. Tilings, C *-algebras, and K-theory (Directions in Mathematical Quasicrystals, 13). American Mathematical Society, Providence, RI, 2000, pp. 177–206; MR 1798993(2001m:46153).Google Scholar
[Len04]Lenz, D.. Aperiodic linearly repetitive Delone sets are densely repetitive. Discrete Comput. Geom. 31(2) (2004), 323–326; MR 2060644(2005a:37027).CrossRefGoogle Scholar
[LMS02]Lee, J. Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5) (2002), 1003–1018; MR 1937612(2004a:52040).CrossRefGoogle Scholar
[LP03]Lagarias, J. C. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23(3) (2003), 831–867; MR 1992666(2005a:52018).CrossRefGoogle Scholar
[LS05]Lenz, D. and Stollmann, P.. An ergodic theorem for Delone dynamical systems and existence of the integrated density of states. J. Anal. Math. 97 (2005), 1–24; MR 2274971(2007m:37020).CrossRefGoogle Scholar
[Moo97]Moody (ed.), R. V.. 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).CrossRefGoogle Scholar
[Pri97]Priebe, N. M.. Detecting hierarchy in tiling dynamical systems via derived Voronoï tessellations. PhD Thesis, University of North Carolina at Chapel Hill, 1997.Google Scholar
[PS01]Priebe, N. and Solomyak, B.. Characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26(3) (2001), 289–306; MR 1854103(2002j:37029).CrossRefGoogle Scholar
[Rob04]Arthur Robinson, E. Jr. Symbolic Dynamics and Tilings of ℝd (Symbolic Dynamics and its Applications, 60). American Mathematical Society, Providence, RI, 2004, pp. 81–119; MR 2078847(2005h:37036).Google Scholar
[SBGC84]Shecthman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long range orientational order and no translational symetry. Phys. Rev. Lett. 53(20) (1984), 1951–1954.Google Scholar
[Sen81]Seneta, E.. Nonnegative Matrices and Markov Chains, 2nd edn(Springer Series in Statistics). Springer, New York, 1981; MR 719544(85i:60058).CrossRefGoogle Scholar
[Sen95]Senechal, M.. Quasicrystals and Geometry. Cambridge University Press, Cambridge, 1995; MR 1340198(96c:52038).Google Scholar
[Sol98]Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2) (1998), 265–279; MR 1637896(99f:52028).Google Scholar
[SW03]Sadun, L. and Williams, R. F.. Tiling spaces are Cantor set fiber bundles. Ergod. Th. & Dynam. Sys. 23(1) (2003), 307–316; MR 1971208(2004a:37023).CrossRefGoogle Scholar