Atomic surfaces, tilings and coincidences II. Reducible case

[Surfaces atomiques, pavages et coïncidences II]

Ei, Hiromi ; Ito, Shunji ; Rao, Hui

Annales de l'Institut Fourier, Tome 56 (2006), p. 2285-2313 / Harvested from Numdam

Résumé

Les surfaces atomiques des substitutions unimodulaires de type Pisot ont été étudiées par de nombreux auteurs. Dans cet article, nous étudions les surfaces atomiques des substitutions Pisot de type réductible.

Par analogie avec le cas irréductible, nous définissons la notion de surfaces plissées et de substitution duale agissant dessus. Grâce à ces notions, nous donnons une preuve simple du fait que les surfaces atomiques forment un système de pavage auto-similaire. Nous montrons que les surfaces atomiques sont quasi-périodiques, ce qui implique qu’un recouvrement non périodique par les surfaces atomiques recouvre l’espace exactement $k$ fois.

Les surfaces atomiques ont été introduites à l’origine par Rauzy dans le but d’étudier le spectre des systèmes dynamiques substitutifs via un pavage périodique. Cependant, nous montrons qu’il n’est pas évident de savoir si les surfaces atomiques peuvent paver l’espace périodiquement ou non, en raison de la complexité du cas réductible. Il semble que la géométrie des surfaces atomiques ne peut pas être appliquée directement au problème spectral.

Abstract

The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.

As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that a non-periodic covering by the atomic surfaces covers the space exactly $k$-times.

The atomic surfaces are originally designed by Rauzy to study the spectrum of the substitution dynamical system via a periodic tiling. However, we show that, since the stepped-surface is complicated in the reducible case, it is not clear whether the atomic surfaces can tile the space periodically or not. It seems that the geometry of the atomic surfaces can not applied directly to the spectral problem.

Détail

Publié le : 2006-01-01

MR 2290782 | Zbl 1119.52013 | 3 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2241
Classification:  52C23,  37A45,  28A80,  11B85
Mots clés: surfaces atomiques, substitution de Pivot, pavages

Bibliographie

[1] Akiyama, S. Self-affine tiling and Pisot numeration system, Number theory and its applications, Kluwer Acad. Plubl., Dordrecht (Dev. Math.) (1999) no. 2, pp. 7-17 | MR 1738803 | Zbl 0999.11065

[2] Akiyama, S. On the boundary of self-affine tilings generated by Pisot numbers, J. Math. Soc. Japan, Tome 54 (2002) no. 2, pp. 283-308 | Article | MR 1883519 | Zbl 1032.11033

[3] Akiyama, S.; Rao, H.; Steiner, W. A certain finiteness property of Pisot number systems, J. Number Theory, Tome 107 (2004), pp. 135-160 | Article | MR 2059954 | Zbl 1052.11055

[4] Arnoux, P.; Berthé, V.; Ito, S. Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble), Tome 52 (2002) no. 2, pp. 305-349 | Article | Numdam | MR 1906478 | Zbl 1017.11006

[5] Arnoux, P.; Ito, S. Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc., Tome 8 (2001), pp. 181-207 | MR 1838930 | Zbl 1007.37001

[6] Baker, V.; Barge, M.; Kwapisz, J. Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts (2005) (Preprint) | MR 2180231

[7] Bandt, C. Self-similar sets. V. Integer matrices and fractal tilings of ${\mathbb{R}}^ n$, Proc. Amer. Math. Soc., Tome 112 (1991) no. 2, pp. 549-562 | MR 1036982 | Zbl 0743.58027

[8] Barge, M.; Diamond, B. Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, Tome 130 (2002) no. 4, pp. 619-626 | Numdam | MR 1947456 | Zbl 1028.37008

[9] Barge, M.; Kwapisz, J. Geometric theory of unimodular Pisot substitutions (2004) (Preprint) | MR 2262174 | Zbl 05071304

[10] Bernat, J.; Berthé, V.; Rao, H. On the super-coincidence condition (2006) (Preprint)

[11] Berthé, V.; Siegel, A. Tilings associated with beta-numeration and substitutions, Electronic J. Comb. Number Theory, Tome 5 (2005) no. 3, pp. #A02 | MR 2191748 | Zbl 05014493

[12] Dekking, F. M. The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Tome 41 (1977/78) no. 3, pp. 221-239 | Article | MR 461470 | Zbl 0348.54034

[13] Durand, F.; Thomas, A. Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., Tome 65 (1989) no. 2, pp. 153-169 | Article | MR 1020484 | Zbl 0679.10010

[14] Ei, H.; Ito, S. Tilings from some non-irreducible, Pisot substitutions, Discrete Mathematics and Theoretical Computer Science, Tome 7 (2005) no. 1, pp. 81-122 | MR 2164061 | Zbl 1153.37323

[15] Ei, H.; Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences III: $\beta $-tiling and super-coincidence (In preparation)

[16] Falconer, K. Techniques in fractal geometry, John Wiley & Sons Ltd., Chichester (1997) | MR 1449135 | Zbl 0869.28003

[17] Fogg, N. Substitutions in dynamics, arithmetics and combinatorics, Springer-Verlag, Berlin, Lecture Notes in Mathematics, 1794 (2002) | MR 1970385 | Zbl 1014.11015

[18] Frougny, C.; Solomyak, B. Finite beta-expansions, Ergodic Theory & Dynam. Sys., Tome 12 (1992), pp. 713-723 | MR 1200339 | Zbl 0814.68065

[19] Host, B. (unpublished manuscript)

[20] Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences I. Irreducible case (To appear in Israel J. Math.) | MR 2254640 | Zbl 1143.37013

[21] Ito, S.; Rao, H. Purely periodic $\beta $-expansions with Pisot unit base, Proc. Amer. Math. Soc., Tome 133 (2005), pp. 953-964 | Article | MR 2117194 | Zbl 02125243

[22] Ito, S.; Sano, Y. On periodic $\beta $-expansions of Pisot numbers and Rauzy fractals, Osaka J. Math., Tome 38 (2001), pp. 349-368 | MR 1833625 | Zbl 0991.11040

[23] Kenyon, R. Self-replicating tilings, P. Walters, ed., Symbolic dynamics and its applications, Contemporary mathematics series, Tome 135 (1992) | Zbl 0770.52013

[24] Lagarias, J.; Wang, Y. Self-affine tiles in ${\mathbb{R}}^ n$, Adv. Math., Tome 121 (1996) no. 1, pp. 21-49 | Article | MR 1399601 | Zbl 0893.52013

[25] Lagarias, J.; Wang, Y. Substitution Delone sets, Discrete Comput. Geom., Tome 29 (2003) no. 2, pp. 175-209 | MR 1957227 | Zbl 1037.52017

[26] Praggastis, B. Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc., Tome 351 (1999) no. 8, pp. 3315-3349 | Article | MR 1615950 | Zbl 0984.11008

[27] Queffelec, M. Substitution Dynamical Systems - Spectral Analysis, Springer-Verlag, Berlin, Lecture Notes in Math., 1294 (1987) | MR 924156 | Zbl 0642.28013

[28] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Tome 110 (1982), pp. 147-178 | Numdam | MR 667748 | Zbl 0522.10032

[29] Senechal, M. Quasicrystals and Geometry, Cambridge University Press (1995) | MR 1340198 | Zbl 0828.52007

[30] Siegel, A. Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, Université de la Méditérranée (2000) (Thèse de doctorat)

[31] Siegel, A. Pure discrete spectrum dynamical system and periodic tiling associated with a substitution, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 2, pp. 341-381 | Article | Numdam | MR 2073838 | Zbl 1083.37009

[32] Sirvent, V.; Wang, Y. Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math., Tome 206 (2002) no. 2, pp. 465-485 | Article | MR 1926787 | Zbl 1048.37015

[33] Thurston, W. P. Groups, tilings, and finite state automata, Providence, RI, AMS Colloquium Lectures (1989)

[34] Thuswaldner, J. Unimodular Pisot substitutions and their associated tiles (To appear in J. Théor. Nombres Bordeaux) | Numdam | Zbl 05135401

[35] Vince, A. Digit tiling of Euclidean space, Center de Recherches Mathematiques CRM Monograph Series, Tome 13 (2000), pp. 329-370 | MR 1798999 | Zbl 0972.52012