zbMATH — the first resource for mathematics

The geometry of non-unit Pisot substitutions. (Géométrie des substitutions de type Pisot non unimodulaires.) (English. French summary) Zbl 1309.05045
G. Rauzy [Bull. Soc. Math. Fr. 110, 147–178 (1982; Zbl 0522.10032)] introduced a family of fractal tiles associated to a specific unit Pisot substitution, and P. Arnoux and S. Ito [Bull. Belg. Math. Soc. - Simon Stevin 8, No. 2, 181–207 (2001; Zbl 1007.37001)] systematically developed these tiling systems. Much of the subsequent developments concentrated on the unit Pisot case until A. Siegel [Ergodic Theory Dyn. Syst. 23, No. 4, 1247–1273 (2003; Zbl 1052.37009)] confirmed part of Rauzy’s conjectures using a representation space including a \(p\)-adic component to build tilings associated to certain Pisot substitutions without the unit assumption. Here, Rauzy crystals are defined and studied in several new situations. The Dumont-Thomas numeration system [J.-M. Dumont and A. Thomas, Theor. Comput. Sci. 65, No. 2, 153–169 (1989; Zbl 0679.10010)] (a generalization of the system of \(\beta\)-numeration) is reviewed, and a system of Rauzy fractals is defined as a natural geometric system associated to the numeration; the geometric realization of a substitution and its dual studied by P. Arnoux and S. Ito [loc. cit.] is extended to the non-unit Pisot case, and constructed as renormalized pieces of stepped hypersurfaces with \(p\)-adic factors; finally B. Sing’s construction [Pisot substitutions and beyond. Bielefeld: Univ. Bielefeld, Fakultät für Mathematik (PhD. Thesis) (2006; Zbl 1210.93006)] of Rauzy crystals using cut-and-project schemes is described using a graph directed iterated function system, in which setting the Rauzy fractals arise as dual prototiles of the associated model set. The different approaches are shown to be related, and the Rauzy fractals are shown to all be the same up to affine equivalence. Here the fractals are subsets of an open subring of the adele ring of the number field defined by the Pisot number. Further dynamical, geometric, and topological properties of Rauzy fractals in these different settings are studied, and the relations between Rauzy crystals and three different structures (subshifts defined using periodic points of the underlying substitution, adic transformations, and domain exchanges of pieces of Rauzy fractals) are discussed.

05B45 Combinatorial aspects of tessellation and tiling problems
11A63 Radix representation; digital problems
11F85 \(p\)-adic theory, local fields
28A80 Fractals
Full Text: DOI arXiv
[1] Akiyama, S., Algebraic number theory and Diophantine analysis (Graz, 1998), Cubic Pisot units with finite beta expansions, 11-26, (2000), de Gruyter, Berlin · Zbl 1001.11038
[2] Akiyama, S., On the boundary of self affine tilings generated by Pisot numbers, J. Math. Soc. Japan, 54, 2, 283-308, (2002) · Zbl 1032.11033
[3] Akiyama, S.; Barat, G.; Berthé, V.; Siegel, A., Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions, Monatsh. Math., 155, 3-4, 377-419, (2008) · Zbl 1190.11005
[4] Arnoux, P.; Ito, S., Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, 8, 2, 181-207, (2001) · Zbl 1007.37001
[5] Baake, M.; Moody, R. V., Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., 573, 61-94, (2004) · Zbl 1188.43008
[6] Barge, M.; Bruin, H.; Jones, L.; Sadun, L., Homological Pisot substitutions and exact regularity, Israel J. Math., 188, 281-300, (2012) · Zbl 1257.37010
[7] Barge, M.; Kwapisz, J., Geometric theory of unimodular Pisot substitutions, Amer. J. Math., 128, 5, 1219-1282, (2006) · Zbl 1152.37011
[8] Berthé, V.; Siegel, A., Tilings associated with beta-numeration and substitutions, Integers, 5, 3, A2, 46 pp., (2005) · Zbl 1139.37008
[9] Berthé, V.; Siegel, A., Purely periodic \(β \)-expansions in the Pisot non-unit case, J. Number Theory, 127, 2, 153-172, (2007) · Zbl 1197.11139
[10] Berthé, V.; Siegel, A.; Steiner, W.; Surer, P.; Thuswaldner, J. M., Fractal tiles associated with shift radix systems, Adv. Math., 226, 1, 139-175, (2011) · Zbl 1221.11018
[11] Berthé, V.; Siegel, A.; Thuswaldner, J., Combinatorics, automata and number theory, 135, Substitutions, Rauzy fractals and tilings, 248-323, (2010), Cambridge Univ. Press, Cambridge · Zbl 1247.37015
[12] Canterini, V.; Siegel, A., Automate des préfixes-suffixes associé à une substitution primitive, J. Théor. Nombres Bordeaux, 13, 2, 353-369, (2001) · Zbl 1071.37011
[13] Canterini, V.; Siegel, A., Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc., 353, 12, 5121-5144, (2001) · Zbl 1142.37302
[14] Dumont, J.-M.; Thomas, A., Systemes de numeration et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., 65, 2, 153-169, (1989) · Zbl 0679.10010
[15] Durand, F., Combinatorics, automata and number theory, 135, Combinatorics on Bratteli diagrams and dynamical systems, 324-372, (2010), Cambridge Univ. Press, Cambridge · Zbl 1272.37006
[16] Fogg, N. P., Substitutions in dynamics, arithmetics and combinatorics, 1794, xviii+402 pp., (2002), Springer-Verlag, Berlin · Zbl 1014.11015
[17] Frougny, C.; Solomyak, B., Finite beta-expansions, Ergodic Theory Dynam. Systems, 12, 4, 713-723, (1992) · Zbl 0814.68065
[18] Ito, S.; Rao, H., Atomic surfaces, tilings and coincidence. I. irreducible case, Israel J. Math., 153, 129-155, (2006) · Zbl 1143.37013
[19] Kalle, C.; Steiner, W., Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc., 364, 5, 2281-2318, (2012) · Zbl 1295.11010
[20] Kuratowski, K., Topology. Vol. I, xx+560 pp., (1966), Academic Press, New York · Zbl 0158.40802
[21] Mauldin, R. D.; Williams, S. C., Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309, 2, 811-829, (1988) · Zbl 0706.28007
[22] Moody, R. V., The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 489, Meyer sets and their duals, 403-441, (1997), Kluwer Acad. Publ., Dordrecht · Zbl 0880.43008
[23] Neukirch, J., Algebraic number theory, 322, xviii+571 pp., (1999), Springer-Verlag, Berlin · Zbl 0956.11021
[24] Queffélec, M., Substitution dynamical systems—spectral analysis, 1294, xvi+351 pp., (2010), Springer-Verlag, Berlin · Zbl 0642.28013
[25] Rauzy, G., Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 2, 147-178, (1982) · Zbl 0522.10032
[26] Rauzy, G., Séminaire de Théorie des Nombres (Talence, 1987-1988), Exp. No. 21, Rotations sur LES groupes, nombres algébriques, et substitutions, (1988), Univ. Bordeaux I, Talence · Zbl 0726.11019
[27] Sano, Y.; Arnoux, P.; Ito, S., Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math., 83, 183-206, (2001) · Zbl 0987.11013
[28] Serre, J.-P., Local fields, 67, viii+241 pp., (1979), Springer-Verlag, New York · Zbl 0423.12016
[29] Siegel, A., Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems, 23, 4, 1247-1273, (2003) · Zbl 1052.37009
[30] Siegel, A.; Thuswaldner, J. M., Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.), 118, 140 pp., (2009) · Zbl 1229.28021
[31] Sing, B., Pisot substitutions and beyond, (2006) · Zbl 1210.93006
[32] Sirvent, V. F.; Solomyak, B., Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Canad. Math. Bull., 45, 4, 697-710, (2002) · Zbl 1038.37008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.