M. S. Akiyama, V. Barge, J. Berthé, A. Lee, and . Siegel, On the Pisot substitution conjecture, Mathematics of aperiodic order, Progress in mathematics, pp.33-72, 2015.

P. Arnoux, V. Berthé, and S. Ito, Plans discrets, actions de ${\Bbb Z}^2$, algorithme de Jacobi-Perron et substitutions, Annales de l???institut Fourier, vol.52, issue.2, pp.305-349, 2002.
DOI : 10.5802/aif.1889

V. [. Avila and . Delecroix, Some monoids of Pisot matrices

B. Adamczewski, Balances for fixed points of primitive substitutions, Theoretical Computer Science, vol.307, issue.1, pp.47-75, 2003.
DOI : 10.1016/S0304-3975(03)00092-6

P. Arnoux and A. M. Fisher, THE SCENERY FLOW FOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING, Chinese Annals of Mathematics, vol.22, issue.04, pp.427-470, 2001.
DOI : 10.1142/S0252959901000425

A. Avila, P. Hubert, and A. Skripchenko, Diffusion for chaotic plane sections of 3-periodic plane surfaces On the Hausdorff dimension of the Rauzy gasket, to appear. [AHS15] to appear. [AI01] P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, pp.181-207, 2000.

J. [. Akiyama and . Lee, Algorithm for determining pure pointedness of self-affine tilings, Advances in Mathematics, vol.226, issue.4, pp.2855-2883, 2011.
DOI : 10.1016/j.aim.2010.07.019

P. Arnoux, M. Mizutani, and T. Sellami, Random product of substitutions with the same incidence matrix, Theoretical Computer Science, vol.543, pp.68-78, 2014.
DOI : 10.1016/j.tcs.2014.06.002

URL : https://hal.archives-ouvertes.fr/hal-01113397

A. [. Arnoux and . Nogueira, Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles, Annales scientifiques de l'??cole normale sup??rieure, vol.26, issue.6, pp.645-664, 1993.
DOI : 10.24033/asens.1682

G. [. Arnoux and . Rauzy, Repr??sentation g??om??trique de suites de complexit?? $2n+1$, Bulletin de la Société mathématique de France, vol.119, issue.2, pp.199-215, 1991.
DOI : 10.24033/bsmf.2164

V. Berthé, J. Bourdon, T. Jolivet, and A. Siegel, Generating discrete planes with substitutions, WORDS, Lecture Notes in Computer Science, vol.8079, pp.119-131, 2013.

V. Berthé, J. Cassaigne, and W. Steiner, BALANCE PROPERTIES OF ARNOUX???RAUZY WORDS, International Journal of Algebra and Computation, vol.23, issue.04, pp.689-703, 2013.
DOI : 10.1142/S0218196713400043

V. [. Berthé and . Delecroix, Beyond substitutive dynamical systems: S-adic expansions, RIMS Lecture note, pp.81-123, 2014.

]. J. Ber07 and . Berstel, Sturmian and episturmian words (a survey of some recent results) Algebraic informatics, Lecture Notes in Comput. Sci, vol.4728, pp.23-47, 2007.

]. V. Ber11, . Berthé-]-v, S. Berthé, L. Q. Ferenczi, and . Zamboni, Multidimensional Euclidean algorithms, numeration and substitutions, Integers 11B Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics, A2. [BFZ05, pp.333-364, 2005.

]. G. Bir57 and . Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc, vol.85, pp.219-227, 1957.

V. Berthé, T. Jolivet, and A. Siegel, Substitutive Arnoux-Rauzy sequences have pure discrete spectrum, Unif. Distrib. Theory, vol.7, issue.1, pp.173-197, 2012.

J. [. Barge and . Kwapisz, Geometric theory of unimodular Pisot substitutions, American Journal of Mathematics, vol.128, issue.5, pp.1219-1282, 2006.
DOI : 10.1353/ajm.2006.0037

V. Berthé, M. Minervino, W. Steiner, and J. Thuswaldner, The S-adic Pisot conjecture on two letters, Top. and its Appl, pp.47-57, 2016.

]. A. Bre81 and . Brentjes, Multidimensional continued fraction algorithms Algorithmes euclidiens pour trois et quatre nombres,Treizì eme congrès des mathèmaticiens scandinaves, tenù a Helsinki 18-23 août, Mathematical Centre Tracts Mathematisch Centrum, vol.145, pp.45-64, 1957.

V. Berthé, A. Siegel, J. M. Thuswaldner, S. Barge, R. F. Stimac et al., Pure discrete spectrum in substitution tiling spaces, Discrete Contin Balance properties of multi-dimensional words, Combinatorics, Automata and Number Theory, Encyclopedia of Mathematics and its Applications Dyn. Syst. Theoret. Comput. Sci, vol.135, issue.33 2 12, pp.579-597, 2002.

J. Cassaigne, S. Ferenczi, and A. Messaoudi, M??lange faible et valeurs propres pour les suites d???Arnoux-Rauzy, Annales de l???institut Fourier, vol.58, issue.6, 1983.
DOI : 10.5802/aif.2403

J. Cassaigne, S. Ferenczi, and L. Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l???institut Fourier, vol.50, issue.4, pp.1265-1276, 2000.
DOI : 10.5802/aif.1792

L. [. Clark and . Sadun, When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam, Systems, vol.23, issue.4, pp.1043-1057, 2003.

]. F. Dek78 and . Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol.4178, issue.3, pp.221-239, 1977.

V. Delecroix, P. Hubert, and S. Evre, Diffusion for the periodic wind-tree model, Annales scientifiques de l'??cole normale sup??rieure, vol.47, issue.6, pp.1085-1110, 2014.
DOI : 10.24033/asens.2234

URL : https://hal.archives-ouvertes.fr/hal-01299347

F. Durand, B. Host, and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam, Systems, vol.19, issue.4, pp.953-993, 1999.

V. Delecroix, T. Hejda, and W. Steiner, Balancedness of Arnoux-Rauzy and Brun Words, Lecture Notes in Computer Science, vol.8079, pp.119-131, 2013.
DOI : 10.1007/978-3-642-40579-2_14

URL : https://hal.archives-ouvertes.fr/hal-00855961

F. Durand, J. Leroy, and G. Richomme, Do the properties of an S-adic representation determine factor complexity?, J. Integer Seq, vol.16, issue.2, 2013.
URL : https://hal.archives-ouvertes.fr/lirmm-00797654

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam, Systems, vol.20, pp.1061-1078, 2000.

]. S. Fer92 and . Ferenczi, Linearly recurrent subshifts have a finite number of nonperiodic subshift factors " [Ergodic Theory Dynam Ergodic Theory Dynam Bounded remainder sets, Acta Arith, Systems Systems, vol.20, issue.23 4, pp.1061-1078, 1992.

T. Fernique, Multidimensional Sturmian sequences and generalized substitutions, Int. J. Found. Comput. Sci, vol.17, pp.575-600, 2006.
URL : https://hal.archives-ouvertes.fr/lirmm-00149363

T. Fujita, S. Ito, M. Keane, and M. Ohtsuki, On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof, Ergodic Theory Dynam, Systems, vol.16, issue.6, pp.1345-1352, 1996.

]. A. Fis09 and . Fisher, Nonstationary mixing and the unique ergodicity of adic transformations, Stoch. Dyn, vol.9, issue.3, pp.335-391, 2009.

H. Furstenberg, H. Keynes, and L. Shapiro, Prime flows in topological dynamics, Israel Journal of Mathematics, vol.15, issue.1, pp.14-26, 1973.
DOI : 10.1007/BF02761532

]. Fog02 and . Fogg, Substitutions in dynamics, arithmetics and combinatorics Frougny and B. Solomyak, Finite beta-expansions, Ergodic Theory Dynam, Lecture Notes in Mathematics Systems, vol.1794, issue.12 4, pp.713-723, 1992.

]. H. Fur60, N. J. Furstenberg, N. Grepstad, and . Lev, Stationary processes and prediction theory Sets of bounded discrepancy for multi-dimensional irrational rotation, Annals of Mathematics Studies Geom. Funct. Anal, issue.44, pp.25-87, 1960.

]. A. Gor07 and . Gorodnik, Open problems in dynamics and related fields, Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals, pp.1-35, 2006.

S. Ito, J. Fujii, H. Higashino, and S. Yasutomi, On simultaneous approximation to (??,??2) with ??3+k?????1=0, Journal of Number Theory, vol.99, issue.2, pp.255-283, 2003.
DOI : 10.1016/S0022-314X(02)00076-8

M. [. Ito and . Ohtsuki, Modified Jacobi-Perron Algorithm and Generating Markov Partitions for Special Hyperbolic Toral Automorphisms, Tokyo Journal of Mathematics, vol.16, issue.2, pp.441-472, 1993.
DOI : 10.3836/tjm/1270128497

P. [. Tilings, J. Algorithm, J. Tokyo, ]. S. Math, H. Ito et al., Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math, vol.17, issue.1, pp.33-58, 1994.

]. S. Ito89 and . Ito, Weyl automorphisms, substitutions and fractals, Stability theory and related topics in dynamical systems, World Sci. Adv. Ser. Dynam. Systems World Sci. Publ, vol.6, pp.60-72, 1988.

S. Ito and S. Yasutomi, On simultaneous Diophantine approximation to periodic points related to modified Jacobi-Perron algorithm, Probability and number theory?Kanazawa, Algorithms, fractals, and dynamics (Okayama/Kyoto Adv. Stud. Pure Math, vol.49, issue.2, pp.95-99, 1992.

]. J. Lag93 and . Lagarias, The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms, Monatsh. Math, vol.115, issue.4, pp.299-328, 1993.

J. [. Labbé and . Leroy, Bispecial Factors in the Brun S-Adic System, Developments in Language Theory (DLT)Mee99] R. Meester, A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm, pp.1077-1083, 1999.
DOI : 10.1007/978-3-662-53132-7_23

J. [. Minervino and . Thuswaldner, G??om??trie des substitutions de type Pisot non unimodulaires, Annales de l???institut Fourier, vol.64, issue.4, pp.1373-1417, 2014.
DOI : 10.5802/aif.2884

]. O. Per07 and . Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann, vol.64, issue.1, pp.1-76, 1907.

N. , P. Frank, and L. Sadun, Fusion: a general framework for hierarchical tilings of R d , Geom, Dedicata, vol.171, pp.149-186, 2014.

]. E. Pod77 and . Podsypanin, A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm Studies in number theory (LOMI), 4. [Que10] M. Queffélec, Substitution dynamical systems?spectral analysis, Rauzy, Nombres algébriques et substitutions, pp.184-194, 1977.

L. [. Risley and . Zamboni, A generalization of Sturmian sequences: combinatorial structure and transcendence, Acta Arith, pp.167-184, 2000.

L. Sadun, Finitely balanced sequences and plasticity of 1-dimensional tilings, Topology Appl, pp.82-87, 2016.

]. B. Sch98 and . Schratzberger, The exponent of convergence for Brun's algorithm in two dimensions

]. B. Sol97 and . Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam, Systems, vol.17, issue.3, pp.695-738, 1997.

A. Siegel and J. M. Thuswaldner, Topological properties of Rauzy fractals Uniform algebraic approximation of shift and multiplication operators, Mém. Soc. Math. Fr. (N.S.) Dokl. Akad. Nauk SSSR, vol.140, issue.259 3, pp.526-529, 1981.

I. and C. Umr, 75205 Paris Cedex 13, FRANCE E-mail address: berthe@liafa.univ-paris-diderot.fr, steiner@liafa.univ-paris-diderot.fr Chair of Mathematics and Statistics