Berthé, Valérie; Jolivet, Timo; Siegel, Anne
Connectedness of fractals associated with Arnoux-Rauzy substitutions
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) no. 3 , p. 249-266
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
MR 3302487
doi : 10.1051/ita/2014008
URL stable : http://www.numdam.org/item?id=ITA_2014__48_3_249_0

Classification:  68R15,  37B10
Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux-Rauzy substitutions.

Bibliographie

[1] B. Adamczewski, C. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic β-expansion. Bull. London Math. Soc. 42 (2010) 538-552. MR 2651949 | Zbl 1211.11010

[2] R.L. Adler, Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. (N.S.) 35 (1998) 1-56. MR 1477538 | Zbl 0892.58019

[3] S. Akiyama and N. Gjini, Connectedness of number theoretic tilings. Discrete Math. Theor. Comput. Sci. 7 (2005) 269-312 (electronic). MR 2183177 | Zbl 1162.11366

[4] S. Akiyama, G. Barat, V. Berthé and A. Siegel, Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. Monatsh. Math. 155 (2008) 377-419. MR 2461585 | Zbl 1190.11005

[5] P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexit*error*é2n + 1. Bull. Soc. Math. France 119 (1991) 199-215. Numdam | MR 1116845 | Zbl 0789.28011

[6] P. Arnoux, V. Berthé, T. Fernique and D. Jamet, Functional stepped surfaces, flips, and generalized substitutions. Theoret. Comput. Sci. 380 (2007) 251-265. MR 2330996 | Zbl 1119.68136

[7] P. Arnoux, V. Berthé and S. Ito, Discrete planes, Z2-actions, Jacobi-Perron algorithm and substitutions. Ann. Inst. Fourier 52 (2002) 305-349. Numdam | MR 1906478 | Zbl 1017.11006

[8] P. Arnoux, V. Berthé and A. Siegel, Two-dimensional iterated morphisms and discrete planes. Theoret. Comput. Sci. 319 (2004) 145-176. MR 2074952 | Zbl 1068.37004

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

[10] M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128 (2006) 1219-1282. MR 2262174 | Zbl 1152.37011

[11] M. Barge, B. Diamond and R. Swanson, The branch locus for one-dimensional Pisot tiling spaces. Fund. Math. 204 (2009) 215-240. MR 2520153 | Zbl 1185.37013

[12] M. Barge, S. Štimac and R.F. Williams, Pure discrete spectrum in substitution tiling spaces. Discrete Contin. Dyn. Syst. 33 (2013) 579-597. MR 2975125 | Zbl 1291.37024

[13] V. Berthé, D. Frettlöh, and V. Sirvent, Selfdual substitutions in dimension one, European J. Combin. 33 (2012) 981-1000. MR 2904970 | Zbl 1252.68164

[14] V. Berthé and M. Rigo, Combinatorics, automata and number theory, Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press (2010). MR 2742574 | Zbl 1197.68006

[15] V. Berthé, S. Ferenczi and L.Q. Zamboni, Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics, Contemp. Math., vol. 385. Amer. Math. Soc. Providence, RI (2005) 333-364. MR 2180244 | Zbl 1156.37301

[16] V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy sequences have pure discrete spectrum. Unif. Distrib. Theory 7 (2012) 173-197. MR 2943167 | Zbl pre06336941

[17] V. Berthé, A. Lacasse, G. Paquin and X. Provençal, A study of Jacobi-Perron boundary words for the generation of discrete planes. Theoret. Comput. Sci. 502 (2013) 118-142. MR 3101696 | Zbl 1296.68113

[18] R. Bowen, Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978) 130-132. MR 474415 | Zbl 0417.58011

[19] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, revised ed., Lect. Notes Math., vol. 470. With a preface by David Ruelle, edited by Jean-René Chazottes. Springer-Verlag, Berlin (2008). MR 2423393 | Zbl 1172.37001

[20] V. Canterini, Connectedness of geometric representation of substitutions of Pisot type. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 77-89. MR 2032327 | Zbl 1031.37015

[21] V. Canterini and A. Siegel, Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353 (2001) 5121-5144. MR 1852097 | Zbl 1142.37302

[22] J. Cassaigne and N. Chekhova, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une question de Morse et Hedlund. Ann. Inst. Fourier Grenoble 56 (2006) 2249-2270. Numdam | MR 2290780 | Zbl 1138.68045

[23] J. Cassaigne, S. Ferenczi and A. Messaoudi, Weak mixing and eigenvalues for Arnoux-Rauzy sequences. Ann. Inst. Fourier 58 (2008) 1983-2005. Numdam | MR 2473626 | Zbl 1151.37013

[24] J. Cassaigne, S. Ferenczi and L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. MR 1799745 | Zbl 1004.37008

[25] H. Ei and S. Ito, Decomposition theorem on invertible substitutions. Osaka J. Math. 35 (1998) 821-834. MR 1659624 | Zbl 0924.20040

[26] T. Fernique, Multidimensional Sturmian sequences and generalized substitutions. Internat. J. Found. Comput. Sci. 17 (2006) 575-599. MR 2234803 | Zbl 1096.68125

[27] T. Fernique, Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 42 (2009) 2229-2238. MR 2503454 | Zbl 1176.68180

[28] J.-P. Gazeau and J.-L. Verger-Gaugry, Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théor. Nombres Bordeaux 16 (2004) 125-149. Numdam | MR 2145576 | Zbl 1075.11007

[29] P. Hubert and A. Messaoudi, Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals. Acta Arith. 124 (2006) 1-15. MR 2262136 | Zbl 1116.28009

[30] S. Ito and M. Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16 (1993) 441-472. MR 1247666 | Zbl 0805.11056

[31] S. Ito and M. Ohtsuki, Parallelogram tilings and Jacobi-Perron algorithm. Tokyo J. Math. 17 (1994) 33-58. MR 1279568 | Zbl 0805.52011

[32] S. Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case. Israel J. Math. 153 (2006) 129-155. MR 2254640 | Zbl 1143.37013

[33] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995). MR 1369092 | Zbl 1106.37301

[34] M. Lothaire, Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997). MR 1475463 | Zbl 0874.20040

[35] A. Messaoudi, Frontière du fractal de Rauzy et système de numération complexe. Acta Arith. 95 (2000) 195-224. MR 1793161 | Zbl 0968.28005

[36] M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1-42. JFM 66.0188.03 | MR 745

[37] B. Praggastis, Numeration systems and Markov partitions from self-similar tilings. Trans. Amer. Math. Soc. 351 (1999) 3315-3349. MR 1615950 | Zbl 0984.11008

[38] N.P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lect. Notes Math., vol. 1794. Springer-Verlag, Berlin (2002). MR 1970385

[39] M. Queffélec, Substitution dynamical systems-spectral analysis, second edition, Lect. Notes Math., vol. 1294. Springer-Verlag, Berlin (2010). MR 2590264 | Zbl 1225.11001

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

[41] J.-P. Reveillès, Géométrie discrète, calculs en nombres entiers et algorithmes, Ph.D. thesis. Université Louis Pasteur, Strasbourg (1991). Zbl 1079.51513

[42] A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type pisot, Ph.D. thesis. Université de la Méditerranée (2000).

[43] A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractal. Mém. Soc. Math. Fr. To appear (2010). Numdam | MR 2721985 | Zbl 1229.28021

[44] B. Tan, Z.-X. Wen and Y. Zhang, The structure of invertible substitutions on a three-letter alphabet. Adv. in Appl. Math. 32 (2004) 736-753. MR 2053843 | Zbl 1082.68092

[45] W. Thurston, Groups, tilings, and finite state automata. AMS Colloquium lecture notes. Unpublished manuscript (1989).