We investigate in a geometrical way the point sets of obtained by the -numeration that are the -integers where is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the -numeration, allowing to lift up the -integers to some points of the lattice ( degree of ) lying about the dominant eigenspace of the companion matrix of . When is in particular a Pisot number, this framework gives another proof of the fact that is a Meyer set. In the internal spaces, the canonical acceptance windows are fractals and one of them is the Rauzy fractal (up to quasi-dilation). We show it on an example. We show that is finitely generated over and make a link with the classification of Delone sets proposed by Lagarias. Finally we give an effective upper bound for the integer taking place in the relation: if (respectively ) has a finite Rényi -expansion.
Nous étudions géométriquement les ensembles de points de obtenus par la -numération que sont les -entiers où est un nombre de Perron. Nous montrons qu’il existe deux schémas de coupe-et-projection canoniques associés à la -numération, où les -entiers se relèvent en certains points du réseau degré de , situés autour du sous-espace propre dominant de la matrice compagnon de . Lorsque est en particulier un nombre de Pisot, nous redonnons une preuve du fait que est un ensemble de Meyer. Dans les espaces internes les fenêtres d’acceptation canoniques sont des fractals dont l’une est le fractal de Rauzy (à quasi-homothétie près). Nous le montrons sur un exemple. Nous montrons que est de type fini sur , faisons le lien avec la classification de Lagarias des ensembles de Delaunay et donnons une borne supérieure effective de l’entier dans la relation : lorsque (respectivement ) a un -développement de Rényi fini.
@article{JTNB_2004__16_1_125_0, author = {Gazeau, Jean-Pierre and Verger-Gaugry, Jean-Louis}, title = {Geometric study of the beta-integers for a {Perron} number and mathematical quasicrystals}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {125--149}, publisher = {Universit\'e Bordeaux 1}, volume = {16}, number = {1}, year = {2004}, doi = {10.5802/jtnb.437}, zbl = {1075.11007}, mrnumber = {2145576}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.437/} }
TY - JOUR AU - Gazeau, Jean-Pierre AU - Verger-Gaugry, Jean-Louis TI - Geometric study of the beta-integers for a Perron number and mathematical quasicrystals JO - Journal de théorie des nombres de Bordeaux PY - 2004 SP - 125 EP - 149 VL - 16 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.437/ DO - 10.5802/jtnb.437 LA - en ID - JTNB_2004__16_1_125_0 ER -
%0 Journal Article %A Gazeau, Jean-Pierre %A Verger-Gaugry, Jean-Louis %T Geometric study of the beta-integers for a Perron number and mathematical quasicrystals %J Journal de théorie des nombres de Bordeaux %D 2004 %P 125-149 %V 16 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.437/ %R 10.5802/jtnb.437 %G en %F JTNB_2004__16_1_125_0
Gazeau, Jean-Pierre; Verger-Gaugry, Jean-Louis. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 125-149. doi : 10.5802/jtnb.437. http://www.numdam.org/articles/10.5802/jtnb.437/
[Ak] S. Akiyama, Cubic Pisot units with finite -expansions. Algebraic number theory and Diophantine analysis, (Graz, 1998), de Gruyter, Berlin, (2000), 11–26. | MR | Zbl
[Ak1] S. Akiyama, Pisot numbers and greedy algorithm. Number Theory, (Eger, 1996), de Gruyter, Berlin, (1998), 9–21. | MR | Zbl
[ABEI] P. Arnoux, V. Berthé, H. Ei and S. Ito, Tilings, quasicrystals, discrete planes, generalized substitutions and multidimensional continued fractions. Disc. Math. and Theor. Comp. Sci. Proc. AA (DM-CCG), (2001), 59–78. | MR | Zbl
[AI] P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. 8 (2001), 181–207. | MR | Zbl
[Be] A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sc. Paris, Série A, t. 285 (1977), 419–421. | MR | Zbl
[Be1] A. Bertrand - Mathis, Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Hung., 54, (3-4) (1989), 237–241. | MR | Zbl
[Be2] A. Bertrand - Mathis, Développement en base , Répartition modulo un de la suite . Langages codes et - shift, Bull. Soc. Math. France, 114 (1986), 271–323. | Numdam | MR | Zbl
[Be3] A. Bertrand - Mathis, Nombres de Perron et questions de rationnalité. Séminaire d’Analyse, Université de Clermont II, Année 1988/89, Exposé 08; A. Bertrand - Mathis, Nombres de Perron et problèmes de rationnalité. Astérisque, 198-199-200 (1991), 67–76. | Numdam | Zbl
[Be4] A. Bertrand - Mathis, Questions diverses relatives aux systèmes codés: applications au -shift. Preprint.
[Bl] F. Blanchard, -expansions and Symbolic Dynamics. Theor. Comp. Sci. 65 (1989), 131–141. | MR | Zbl
[Bu] Č. Burdík, Ch. Frougny, J.-P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A: Math. Gen. 31 (1998), 6449–6472. | MR | Zbl
[Fro] C. Frougny, Number representation and finite automata. London Math. Soc. Lecture Note Ser. 279 (2000), 207–228. | MR | Zbl
[Fro1] Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergod. Theor. Dynam. Syst. 12 (1992), 713–723. | MR | Zbl
[Gaz] J. P. Gazeau, Pisot-Cyclotomic Integers for Quasicrystals. The Mathematics of Long-Range Aperiodic Order, Ed. By R.V. Moody, Kluwer Academic Publishers, (1997), 175–198. | MR | Zbl
[HS] M. W. Hirsch and S. Smale, Differential Equations. Dynamical Systems and Linear Algebra, Academic Press, New York, (1974). | MR | Zbl
[IK] S. Ito and M. Kimura, On the Rauzy fractal. Japan J. Indust. Appl. Math. 8 (1991), 461–486. | MR | Zbl
[IS] S. Ito and Y. Sano, On periodic - expansions of Pisot numbers and Rauzy fractals. Osaka J. Math. 38 (2001), 349–368. | MR | Zbl
[La] J. Lagarias, Geometric Models for Quasicrystals I. Delone Sets of Finite Type. Discrete Comput. Geom. 21 no 2 (1999), 161–191. | MR | Zbl
[La1] J.C. Lagarias, Geometrical models for quasicrystals. II. Local rules under isometry. Disc. and Comp. Geom., 21 no 3 (1999), 345–372. | MR | Zbl
[La2] J.C. Lagarias, Mathematical Quasicrystals and the problem of diffraction. Directions in Mathematical Physics, CRM Monograph Series, Vol 13, Ed. M. Baake and R.V. Moody, (2000), 61–94. | MR | Zbl
[Li] D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4 (1984), 283–300. | MR | Zbl
[Me] A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy. J. Théor. Nombres Bordeaux, 10 (1998), 135–162. | EuDML | Numdam | MR | Zbl
[Me1] A. Messaoudi, Frontière du fractal de Rauzy et système de numérotation complexe Acta Arith. 95 (2000), 195–224. | EuDML | MR | Zbl
[Mey] Y. Meyer, Algebraic Numbers and Harmonic Analysis. North-Holland (1972). | MR | Zbl
[Mi] H. Minc, Nonnegative matrices. John Wiley and Sons, New York (1988). | MR | Zbl
[Mo] R.V. Moody, Meyer sets and their duals The Mathematics of Long-Range Aperiodic Order, Ed. By R.V. Moody, Kluwer Academic Publishers, (1997), 403–441. | MR | Zbl
[MVG] G. Muraz and J.-L. Verger-Gaugry, On lower bounds of the density of packings of equal spheres of . Institut Fourier, preprint n 580 (2003).
[Pa] W. Parry, On the - expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960), 401–416. | MR | Zbl
[PF] N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lect. Notes Math. 1794, Springer-Verlag, (2003). | MR | Zbl
[Ra] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France, 110 (1982), 147–178. | EuDML | Numdam | MR | Zbl
[Re] A. Renyi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477–493. | MR | Zbl
[Ru] D. Ruelle, Statistical Mechanics; Rigorous results. Benjamin, New York, (1969). | MR | Zbl
[Sc] J. Schmeling, Symbolic dynamics for - shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694. | MR | Zbl
[Sch] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278. | MR | Zbl
[So] B. Solomyak, Dynamics of Self-Similar Tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738. | MR | Zbl
[Th] W. Thurston Groups, tilings and finite state automata. Preprint (1989).
Cited by Sources: