Skip to main content
×
×
Home

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.

Copyright
References
Hide All
[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), 509537; MR 1631708(2000a:46112).
[BBG06]Bellissard J., Benedetti R. and Gambaudo J. M.. Spaces of tilings, finite telescopic approximations and gap-labeling. Comm. Math. Phys. 261(1) (2006), 141; MR 2193205(2007c:46063).
[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), 799816; MR 2190338(2006j:37011).
[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), 639664.
[Bes08a]Besbes A.. Uniform ergodic theorems on aperiodic linearly repetitive tilings and applications. Rev. Math. Phys. 20(5) (2008), 597623; MR 2422207.
[Bes08b]Besbes A.. Contributions a l’étude de quelques systèmes quasi-crystallographics (in French). PhD Thesis, Université Pierre et Marie Curie, Paris, 2008.
[BG03]Benedetti R. and Gambaudo J. M.. On the dynamics of 𝔾-solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23(3) (2003), 673691; MR 1992658(2004f:37019).
[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), 790804; MR 1967706(2004b:37018).
[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).
[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), 23052315; MR 2276621(2007k:37010).
[Cor]Coronel D.. The cohomological equation over dynamical systems arising from Delone sets. Ergod. Th. & Dynam. Sys., doi:10.1017/S0143385710000209.
[DL06]Damanik D. and Lenz D.. Substitution dynamical systems: characterization of linear repetitivity and applications. J. Math. Anal. Appl. 321(2) (2006), 766780; MR 2241154(2007d:37008).
[Dur00]Durand F.. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys. 20(4) (2000), 10611078; MR 1779393(2001m:37022).
[For00]Forrest A.. A Bratteli diagram for commuting homeomorphisms of the Cantor set. Internat. J. Math. 11(2) (2000), 177200; MR 1754619(2001d:37008).
[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).
[GM06]Gambaudo J.-M. and Martens M.. Algebraic topology for minimal Cantor sets. Ann. Henri Poincaré 7(3) (2006), 423446; MR 2226743(2006m:37007).
[GMPS10]Giordano T., Matui H., Putnam I. F. and Skau C. F.. Orbit equivalence for Cantor minimal ℤd-systems. Invent. Math. 179(1) (2010), 119158.
[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), 827864; MR 1194074(94f:46096).
[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. 177206; MR 1798993(2001m:46153).
[Len04]Lenz D.. Aperiodic linearly repetitive Delone sets are densely repetitive. Discrete Comput. Geom. 31(2) (2004), 323326; MR 2060644(2005a:37027).
[LMS02]Lee J. Y., Moody R. V. and Solomyak B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5) (2002), 10031018; MR 1937612(2004a:52040).
[LP03]Lagarias J. C. and Pleasants P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23(3) (2003), 831867; MR 1992666(2005a:52018).
[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), 124; MR 2274971(2007m:37020).
[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).
[Pri97]Priebe N. M.. Detecting hierarchy in tiling dynamical systems via derived Voronoï tessellations. PhD Thesis, University of North Carolina at Chapel Hill, 1997.
[PS01]Priebe N. and Solomyak B.. Characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26(3) (2001), 289306; MR 1854103(2002j:37029).
[Rob04]Arthur Robinson E. Jr. Symbolic Dynamics and Tilings of ℝd (Symbolic Dynamics and its Applications, 60). American Mathematical Society, Providence, RI, 2004, pp. 81119; MR 2078847(2005h:37036).
[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), 19511954.
[Sen81]Seneta E.. Nonnegative Matrices and Markov Chains, 2nd edn(Springer Series in Statistics). Springer, New York, 1981; MR 719544(85i:60058).
[Sen95]Senechal M.. Quasicrystals and Geometry. Cambridge University Press, Cambridge, 1995; MR 1340198(96c:52038).
[Sol98]Solomyak B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2) (1998), 265279; MR 1637896(99f:52028).
[SW03]Sadun L. and Williams R. F.. Tiling spaces are Cantor set fiber bundles. Ergod. Th. & Dynam. Sys. 23(1) (2003), 307316; MR 1971208(2004a:37023).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 273 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th January 2018. This data will be updated every 24 hours.