The geometry of non-unit Pisot substitutions

Minervino, Milton; Thuswaldner, Jörg

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1373-1417.

On peut associer à une substitution de type Pisot non unimodulaire $\sigma$ certaines tuiles fractales, appelés fractals de Rauzy. Dans ce contexte, ces fractals sont des sous-ensembles d’un certain sous-anneau ouvert de l’anneau des adèles du corps de nombres associé. On présente plusieurs approches sur la façon de définir les fractals de Rauzy. En particulier, on considère les fractals de Rauzy comme des objets géométriques naturels associés à certains systèmes de numération, en termes du dual de la réalisation unidimensionnelle de $\sigma$ et comme des ensembles définis par coupe et projection. On définit également des surfaces discrètes adaptées aux substitutions de type Pisot non unimodulaires. On établit des propriétés topologiques et géométriques basiques des fractals de Rauzy, ainsi que des résultats de pavage. Finalement on fournit des relations entre des sous-décalages définis en termes de points périodiques de $\sigma$, des transformations adiques et un échange de morceaux.

It is known that with a non-unit Pisot substitution $\sigma$ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of $\sigma$, and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma$, to adic transformations, and a domain exchange.

Classification : 05B45, 11A63, 11F85, 28A80
Keywords: Rauzy fractal, tiling, $p$-adic completion, beta-numeration
Mot clés : fractals de Rauzy, pavage, complété $p-$-adique, beta-numération

[1] Akiyama, S. Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter, Berlin, 2000, pp. 11-26 | MR | Zbl

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

[3] Akiyama, S.; Barat, G.; Berthé, V.; Siegel, A. Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions, Monatsh. Math., Volume 155 (2008) no. 3-4, pp. 377-419 | DOI | MR | Zbl

[4] Arnoux, P.; Ito, S. Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Volume 8 (2001) no. 2, pp. 181-207 Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000) | MR | Zbl

[5] Baake, M.; Moody, R. V. Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., Volume 573 (2004), pp. 61-94 | MR | Zbl

[6] Barge, M.; Bruin, H.; Jones, L.; Sadun, L. Homological Pisot substitutions and exact regularity, Israel J. Math., Volume 188 (2012), pp. 281-300 | DOI | MR | Zbl

[7] Barge, M.; Kwapisz, J. Geometric theory of unimodular Pisot substitutions, Amer. J. Math., Volume 128 (2006) no. 5, pp. 1219-1282 | DOI | MR | Zbl

[8] Berthé, V.; Siegel, A. Tilings associated with beta-numeration and substitutions, Integers, Volume 5 (2005) no. 3, pp. A2, 46 | EuDML | MR | Zbl

[9] Berthé, V.; Siegel, A. Purely periodic $\beta$-expansions in the Pisot non-unit case, J. Number Theory, Volume 127 (2007) no. 2, pp. 153-172 | DOI | MR | Zbl

[10] Berthé, V.; Siegel, A.; Steiner, W.; Surer, P.; Thuswaldner, J. M. Fractal tiles associated with shift radix systems, Adv. Math., Volume 226 (2011) no. 1, pp. 139-175 | DOI | MR | Zbl

[11] Berthé, V.; Siegel, A.; Thuswaldner, J. Substitutions, Rauzy fractals and tilings, Combinatorics, automata and number theory (Encyclopedia Math. Appl.), Volume 135, Cambridge Univ. Press, Cambridge, 2010, pp. 248-323 | MR | Zbl

[12] Canterini, V.; Siegel, A. Automate des préfixes-suffixes associé à une substitution primitive, J. Théor. Nombres Bordeaux, Volume 13 (2001) no. 2, pp. 353-369 | DOI | Numdam | MR | Zbl

[13] Canterini, V.; Siegel, A. Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc., Volume 353 (2001) no. 12, pp. 5121-5144 | DOI | MR | Zbl

[14] Dumont, J.-M.; Thomas, A. Systemes de numeration et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., Volume 65 (1989) no. 2, pp. 153-169 | DOI | MR | Zbl

[15] Durand, F. Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, automata and number theory (Encyclopedia Math. Appl.), Volume 135, Cambridge Univ. Press, Cambridge, 2010, pp. 324-372 | MR | Zbl

[16] Fogg, N. P. Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002, pp. xviii+402 (Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel) | MR

[17] Frougny, C.; Solomyak, B. Finite beta-expansions, Ergodic Theory Dynam. Systems, Volume 12 (1992) no. 4, pp. 713-723 | DOI | MR | Zbl

[18] Ito, S.; Rao, H. Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math., Volume 153 (2006), pp. 129-155 | DOI | MR | Zbl

[19] Kalle, C.; Steiner, W. Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc., Volume 364 (2012) no. 5, pp. 2281-2318 | DOI | MR | Zbl

[20] Kuratowski, K. Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York, 1966, pp. xx+560 | MR | Zbl

[21] Mauldin, R. D.; Williams, S. C. Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., Volume 309 (1988) no. 2, pp. 811-829 | DOI | MR | Zbl

[22] Moody, R. V. Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403-441 | MR | Zbl

[23] Neukirch, J. Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, Springer-Verlag, Berlin, 1999, pp. xviii+571 (Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder) | MR | Zbl

[24] Queffélec, M. Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 2010, pp. xvi+351 | MR | Zbl

[25] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Volume 110 (1982) no. 2, pp. 147-178 | Numdam | MR | Zbl

[26] Rauzy, G. Rotations sur les groupes, nombres algébriques, et substitutions, Séminaire de Théorie des Nombres (Talence, 1987-1988), Exp. No. 21, Univ. Bordeaux I, Talence, 1988 | Zbl

[27] Sano, Y.; Arnoux, P.; Ito, S. Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math., Volume 83 (2001), pp. 183-206 | DOI | MR | Zbl

[28] Serre, J.-P. Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York, 1979, pp. viii+241 (Translated from the French by Marvin Jay Greenberg) | MR | Zbl

[29] Siegel, A. Représentation des systèmes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems, Volume 23 (2003) no. 4, pp. 1247-1273 | DOI | MR | Zbl

[30] Siegel, A.; Thuswaldner, J. M. Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.) (2009) no. 118, pp. 140 | Numdam | MR | Zbl

[31] Sing, B. Pisot substitutions and beyond, University of Bielefeld (2006) http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:hbz:361-11555 (Ph. D. Thesis) | Zbl

[32] Sirvent, V. F.; Solomyak, B. Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Canad. Math. Bull., Volume 45 (2002) no. 4, pp. 697-710 (Dedicated to Robert V. Moody) | DOI | MR | Zbl