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Kolakoski-(3, 1) Is a (Deformed) Model Set

Published online by Cambridge University Press:  20 November 2018

Michael Baake  and
Bernd Sing
Bernd Sing
Affiliation:
Institut für Mathematik Universität Greifswald Jahnstr. 15a 17487 Greifswald Germany, e-mail: sing@uni-greifswald.de, http://schubert.math-inf.uni-greifswald.de
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Abstract

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Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.

Keywords

52C2337B1028A8043A25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 2 , 01 June 2004 , pp. 168 - 190
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Arnoux, P. and Ito, S., Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181–207.Google Scholar
[2] Baake, M., Diffraction of weighted lattice subsets. Canad. Math. Bull. 45 (2002), 483–498. math.MG/0106111Google Scholar
[3] Baake, M., A guide to mathematical quasicrystals. In: Quasicrystals, (eds., J.-B. Suck, M. Schreiber and P. Häussler), Springer, Berlin, 2002, 17–48. math-ph/9901014Google Scholar
[4] Baake, M. and Lenz, D., Deformation of Delone dynamical systems and topological conjugacy. Preprint (2003).Google Scholar
[5] Baake, M. and Moody, R. V., Self-similar measures for quasicrystals. In: Directions in Mathematical Quasicrystals, (eds., M. Baake and R. V. Moody), Amer.Math. Soc., Providence, 2000, 1–42. math.MG/0008063Google Scholar
[6] Baake, M. and Moody, R. V., Weighted Dirac combs with pure point diffraction. To appear in: J. Reine Angew.Math. math.MG/0203030Google Scholar
[7] Baake, M., Moody, R. V. and Schlottmann, M., Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces. J. Phys. A 31 (1998), 5755–5765. math-ph/9901008Google Scholar
[8] Bandt, C., Self-similar tilings and patterns described by mappings. In: The Mathematics of Long-Range Aperiodic Order, (ed., R. V. Moody), Kluwer, Dordrecht (1997), 45–83.Google Scholar
[9] Bernuau, G. and Duneau, M., Fourier Analysis of deformed model sets. In: Directions in Mathematical Quasicrystals, (eds., M. Baake and R. V. Moody), AMS, Providence (2000), 43–60.Google Scholar
[10] Bombieri, E. and Taylor, J. E., Which distributions of matter diffract? An initial investigation. J. Physique Coll. C 3 (1986), 19–29.Google Scholar
[11] Borewicz, S. I. and Šafarevič, I. R., Zahlentheorie. Birkhäuser, Basel, 1966.Google Scholar
[12] Canterini, V. and Siegel, A., Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353 (2001), 5121–5144.Google Scholar
[13] Dekking, F. M., The spectrum of dynamical systems arising from substitutions of constant length. Z.Wahrsch. Verw. Gebiete 41 (1978), 221–239.Google Scholar
[14] Dekking, F. M., Regularity and irregularity of sequences generated by automata. Sém. Th. Nombres Bordeaux 1979–80, exposé 9, 901–910.Google Scholar
[15] Dekking, F. M., What is the long range order in the Kolakoski sequence? In: The Mathematics of Long-Range Aperiodic Order, (ed., R. V. Moody), Kluwer, Dordrecht, 1997, 115–125.Google Scholar
[16] Edgar, G. A.,Measure, Topology and Fractal Geometry. Springer, New York, 1990.Google Scholar
[17] Gähler, F. and Klitzing, R., The diffraction pattern of self-similar tilings. In: The Mathematics of Long-Range Aperiodic Order, (ed., R. V. Moody), Kluwer, Dordrecht, 1997, 141–174.Google Scholar
[18] Hof, A., On diffraction by aperiodic structures. Comm. Math. Phys. 169 (1995), 25–43.Google Scholar
[19] Hof, A., Diffraction by aperiodic structures. In: The Mathematics of Long-Range Aperiodic Order, (ed., R. V. Moody), Kluwer, Dordrecht, 1997, 239–268.Google Scholar
[20] Hutchinson, J. E., Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
[21] Kolakoski, W., Self generating runs, Problem 5304. Amer. Math.Monthly 72(1965), 674.Google Scholar
[22] Lee, J.-Y. and Moody, R. V., Lattice substitution systems and model sets. Discrete Comput. Geom. 25 (2001), 173–201. math.MG/0002019Google Scholar
[23] Lee, J.-Y., Moody, R. V. and Solomyak, B., Pure point dynamical and diffraction spectra. Ann. Inst. H. Poincaré 3 (2002), 1003–1018. mp arc/02-39Google Scholar
[24] Luck, J. M., Godrèche, C., Janner, A. and Janssen, T., The nature of the atomic surfaces of quasiperiodic self-similar structures. J. Phys. A 26 (1993), 1951–1999.Google Scholar
[25] Moody, R. V., Meyer sets and their duals. In: The Mathematics of Long-Range Aperiodic Order, (ed., R. V. Moody), Kluwer, Dordrecht, 1997, 403–441.Google Scholar
[26] Moody, R. V., Model sets: a survey. In: From Quasicrystals to More Complex Systems, (eds., F. Axel, F. Dénoyer and J.P. Gazeau), EDP Sciences, Les Ulis, and Springer, Berlin, 2000, 145–166. math.MG/0002020Google Scholar
[27] Schlottmann, M., Geometrische Eigenschaften quasiperiodischer Strukturen. Dissertation, Universität Tübingen, 1993.Google Scholar
[28] Schlottmann, M., Cut-and-project sets in locally compact Abelian groups. In: Quasicrystals and Discrete Geometry, (ed., J. Patera), Amer.Math. Soc., Providence, 1998, 247–264.Google Scholar
[29] Schlottmann, M., Generalized model sets and dynamical systems. In: Directions in Mathematical Quasicrystals, (eds., M. Baake and R. V. Moody), Amer.Math. Soc., Providence, 2000, 43–60.Google Scholar
[30] Siegel, A., Represéntation des systèmes dynamiques substitutifs non unimodulaires. Ergodic Theory Dynam. Systems 23 (2003), 1247–1273.Google Scholar
[31] Sing, B., Spektrale Eigenschaften der Kolakoski-Sequenzen. Diploma Thesis, Universität Tübingen, 2002, available from the author.Google Scholar
[32] Sing, B., Kolakoski-(2m, 2n) are limit-periodic model sets. J. Math. Phys. 44(2003), 899–912. math-ph/0207037Google Scholar
[33] Sirvent, V. F., Modélos geométricos asociados a substituciones. Habilitation (trabajo de ascenso), Universidad Simón Bolívar (1998); available at: http://www.ma.usb.ve/~vsirvent/publi.html. Google Scholar
[34] Sirvent, V. F. and Solomyak, B., Pure discrete spectrum for one-dimensional substitutions of Pisot type. Canad. Math. Bull. 45 (2002), 697–710.Google Scholar
[35] Sirvent, V. F. and Wang, Y., Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206 (2002), 465–485.Google Scholar