Geometry, dynamics, and arithmetic of $S$-adic shifts

Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jörg M.

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1347-1409.

Cet article étudie les propriétés géométriques et spectrales de décalages $S$-adiques engendrés par fractions continues. Ces systèmes dynamiques symboliques sont obtenus par itération adique d’une suite de substitutions. Nous montrons que ces décalages sont à spectre purement discret et étudions les propriétés des ensembles fractals de Rauzy associés sous une hypothèse de type Pisot généralisée ainsi qu’une condition géométrique de coïncidence. Ces résultats étendent la portée de la conjecture Pisot substitutive au cadre $S$-adique. Nous montrons que presque tous les décalages d’Arnoux–Rauzy ont un spectre purement discret. En utilisant des mots $S$-adiques liés à l’algorithme de fraction continue de Brun, nous exhibons des ensembles à restes bornés et des codages symboliques pour presque toutes les translations du tore bidimensionnel. En raison de l’absence des propriétés d’autosimilarité des systèmes substitutifs, nous devons développer de nouvelles preuves dans le cadre $S$-adique.

This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux–Rauzy as well as Brun substitutions. It is shown that almost all of these shifts have pure discrete spectrum. Using $S$-adic words related to Brun’s continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus. Due to the lack of self-similarity properties present for substitutive systems we have to develop new proofs to obtain our results in the $S$-adic setting.

Classification : 37B10, 37A30, 11K50, 28A80
Keywords: Symbolic dynamics, non-stationary dynamics, $S$-adic shifts, substitutions, tilings, Pisot numbers, continued fractions, Brun algorithm, Arnoux–Rauzy algorithm, Lyapunov exponents
Mot clés : Dynamique symbolique, dynamique non stationnaire, décalages $S$-adiques, substitutions, pavages, nombres de Pisot, fractions continues, algorithme de Brun, algorithme d’Arnoux–Rauzy, exposants de Lyapunov

[1] Adamczewski, Boris Balances for fixed points of primitive substitutions, Theor. Comput. Sci., Volume 307 (2003) no. 1, pp. 47-75 | MR | Zbl

[2] Adamczewski, Boris Symbolic discrepancy and self-similar dynamics, Ann. Inst. Fourier, Volume 54 (2004) no. 7, pp. 2201-2234 | DOI | MR | Zbl

[3] Akiyama, Shigeki; Barge, Marcy; Berthé, Valérie; Lee, Jeong-Yup; Siegel, Anne On the Pisot substitution conjecture, Mathematics of aperiodic order (Progress in Mathematics), Volume 309, Birkhäuser, 2015, pp. 33-72 | DOI | MR

[4] Akiyama, Shigeki; Lee, Jeong-Yup Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., Volume 226 (2011) no. 4, pp. 2855-2883 | DOI | MR | Zbl

[5] Arnoux, Pierre; Berthé, Valérie; Ito, Shunji Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier, Volume 52 (2002) no. 2, pp. 305-349 | DOI | MR | Zbl

[6] Arnoux, Pierre; Berthé, Valérie; Minervino, Milton; Steiner, Wolfgang; Thuswaldner, Jörg M. Nonstationary Markov Partitions, Flows on Homogeneous Spaces, and Generalized Continued Fractions (2018) (in preparation)

[7] Arnoux, Pierre; Fisher, Albert M. The scenery flow for geometric structures on the torus: the linear setting, Chin. Ann. Math., Ser. B, Volume 22 (2001) no. 4, pp. 427-470 | DOI | MR | Zbl

[8] Arnoux, Pierre; Fisher, Albert M. Anosov families, renormalization and non-stationary subshifts, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 3, pp. 661-709 | DOI | MR | Zbl

[9] Arnoux, Pierre; Ito, Shunji Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Volume 8 (2001) no. 2, pp. 181-207 | MR | Zbl

[10] Arnoux, Pierre; Mizutani, Masahiro; Sellami, Tarek Random product of substitutions with the same incidence matrix, Theor. Comput. Sci., Volume 543 (2014), pp. 68-78 | DOI | MR | Zbl

[11] Arnoux, Pierre; Nogueira, Arnaldo Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 6, pp. 645-664 | DOI | MR | Zbl

[12] Arnoux, Pierre; Rauzy, Gérard Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. Fr., Volume 119 (1991) no. 2, pp. 199-215 | DOI | Zbl

[13] Avila, Artur; Delecroix, Vincent Some monoids of Pisot matrices (2015) (https://arxiv.org/abs/1506.03692)

[14] Avila, Artur; Hubert, Pascal; Skripchenko, Alexandra Diffusion for chaotic plane sections of $3$-periodic plane surfaces, Invent. Math., Volume 206 (2016), pp. 109-146 | DOI | MR | Zbl

[15] Avila, Artur; Hubert, Pascal; Skripchenko, Alexandra On the Hausdorff dimension of the Rauzy gasket, Bull. Soc. Math. Fr., Volume 144 (2016), pp. 539-568 | DOI | MR | Zbl

[16] Barge, Marcy Pure discrete spectrum for a class of one-dimensional substitution tiling systems, Discrete Contin. Dyn. Syst., Volume 36 (2016), pp. 1159-1173 | DOI | MR | Zbl

[17] Barge, Marcy The Pisot conjecture for $\beta$-substitutions, Ergodic Theory Dyn. Syst., Volume 38 (2018), pp. 444-472 | DOI | MR | Zbl

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

[19] Barge, Marcy; Štimac, Sonja; Williams, Robert F. Pure discrete spectrum in substitution tiling spaces, Discrete Contin. Dyn. Syst., Volume 33 (2013) no. 2, pp. 579-597 | MR | Zbl

[20] Berstel, Jean Sturmian and episturmian words (a survey of some recent results), Algebraic informatics (Lecture Notes in Computer Science), Volume 4728, Springer, 2007, pp. 23-47 | DOI | MR | Zbl

[21] Berthé, Valérie Multidimensional Euclidean algorithms, numeration and substitutions, Integers, Volume 11B (2011), A02, 34 pages (Art. ID A02, 34 pages) | MR | Zbl

[22] Berthé, Valérie; Bourdon, Jérémie; Jolivet, Timo; Siegel, Anne Generating Discrete Planes with Substitutions, Combinatorics on words. 9th international conference, WORDS 2013 (Lecture Notes in Computer Science), Volume 8079, Springer (2013), pp. 58-70 | MR | Zbl

[23] Berthé, Valérie; Bourdon, Jérémie; Jolivet, Timo; Siegel, Anne A combinatorial approach to products of Pisot substitutions, Ergodic Theory Dyn. Syst. (2015), pp. 1-38 | Zbl

[24] Berthé, Valérie; Cassaigne, Julien; Steiner, Wolfgang Balance properties of Arnoux-Rauzy words, Int. J. Algebra Comput., Volume 23 (2013) no. 4, pp. 689-703 | DOI | MR | Zbl

[25] Berthé, Valérie; Delecroix, Vincent Beyond substitutive dynamical systems: $S$-adic expansions, RIMS Kôkyûroku Bessatsu, Volume B46 (2014), pp. 81-123 | Zbl

[26] Berthé, Valérie; Ferenczi, Sébastien; Zamboni, Luca Q. Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics (Contemporary Mathematics), Volume 385, American Mathematical Society, 2005, pp. 333-364 | DOI | MR | Zbl

[27] Berthé, Valérie; Jolivet, Timo; Siegel, Anne Substitutive Arnoux-Rauzy sequences have pure discrete spectrum, Unif. Distrib. Theory, Volume 7 (2012) no. 1, pp. 173-197 | MR | Zbl

[28] Berthé, Valérie; Minervino, Milton; Steiner, Wolfgang; Thuswaldner, Jörg M. The $S$-adic Pisot conjecture on two letters, Topology Appl., Volume 205 (2016), pp. 47-57 | DOI | MR | Zbl

[29] Berthé, Valérie; Siegel, Anne; Thuswaldner, Jörg M. Substitutions, Rauzy fractals, and tilings, Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and Its Applications), Volume 135, Cambridge University Press, 2010 | DOI | MR | Zbl

[30] Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jörg M.; Yassawi, Reem Recognizability for sequences of morphisms, Ergodic Theory Dyn. Syst. (2018) | DOI

[31] Berthé, Valérie; Tijdeman, Robert Balance properties of multi-dimensional words, Theor. Comput. Sci., Volume 273 (2002) no. 1-2, pp. 197-224 | DOI | MR | Zbl

[32] Birkhoff, Garrett Extensions of Jentzsch’s theorem, Trans. Am. Math. Soc., Volume 85 (1957), pp. 219-227 | MR | Zbl

[33] Brentjes, Arne J. Multidimensional continued fraction algorithms, Mathematical Centre Tracts, 145, Mathematisch Centrum, 1981, i+183 pages | MR | Zbl

[34] Broise-Alamichel, Anne On the characteristic exponents of the Jacobi-Perron algorithm, Dynamical systems and Diophantine approximation (Séminaires et Congrès), Volume 19, Société Mathématique de France, 2009, pp. 151-171 | MR | Zbl

[35] Brun, Viggo Algorithmes euclidiens pour trois et quatre nombres, Treizième congrès des mathèmaticiens scandinaves, tenu à Helsinki 18-23 août 1957, Mercators Tryckeri, 1958, pp. 45-64 | MR | Zbl

[36] Cassaigne, Julien; Ferenczi, Sébastien; Messaoudi, Ali Weak mixing and eigenvalues for Arnoux–Rauzy sequences, Ann. Inst. Fourier, Volume 58 (2008) no. 6, pp. 1983-2005 | DOI | MR | Zbl

[37] Cassaigne, Julien; Ferenczi, Sébastien; Zamboni, Luca Q. Imbalances in Arnoux–Rauzy sequences, Ann. Inst. Fourier, Volume 50 (2000) no. 4, pp. 1265-1276 | DOI | MR | Zbl

[38] Chevallier, Nicolas Coding of a translation of the two-dimensional torus, Monatsh. Math., Volume 157 (2009) no. 2, pp. 101-130 | DOI | MR | Zbl

[39] Clark, Alex; Sadun, Lorenzo When size matters: subshifts and their related tiling spaces, Ergodic Theory Dyn. Syst., Volume 23 (2003) no. 4, pp. 1043-1057 | DOI | MR | Zbl

[40] Dekking, Frederik M. The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 41 (1978) no. 3, pp. 221-239 | DOI | MR | Zbl

[41] Delecroix, Vincent; Hejda, Tomáš; Steiner, Wolfgang Balancedness of Arnoux-Rauzy and Brun Words, WORDS (Lecture Notes in Computer Science), Volume 8079, Springer (2013), pp. 119-131 | MR | Zbl

[42] Delecroix, Vincent; Hubert, Pascal; Lelièvre, S. Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 6, pp. 1085-1110 | DOI | MR | Zbl

[43] Durand, Fabien Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dyn. Syst., Volume 20 (2000), pp. 1061-1078 | DOI | MR | Zbl

[44] Durand, Fabien Corrigendum and addendum to: “Linearly recurrent subshifts have a finite number of non-periodic subshift factors” [Ergodic Theory Dynam. Systems 20 (2000), 1061–1078], Ergodic Theory Dyn. Syst., Volume 23 (2003), pp. 663-669 | MR

[45] Durand, Fabien; Host, Bernard; Skau, Christian Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 4, pp. 953-993 | DOI | MR | Zbl

[46] Durand, Fabien; Leroy, Julien; Richomme, Gwenaël Do the properties of an $S$-adic representation determine factor complexity?, J. Integer Seq., Volume 16 (2013) no. 2, 13.2.6, 30 pages (Art. ID 13.2.6, 30 pages) | MR | Zbl

[47] Ferenczi, Sébastien Bounded remainder sets, Acta Arith., Volume 61 (1992) no. 4, pp. 319-326 | DOI | MR | Zbl

[48] Fernique, Thomas Multidimensional Sturmian sequences and generalized substitutions, Int. J. Found. Comput. Sci., Volume 17 (2006) no. 3, pp. 575-600 | DOI | MR | Zbl

[49] Fisher, Albert M. Nonstationary mixing and the unique ergodicity of adic transformations, Stoch. Dyn., Volume 9 (2009) no. 3, pp. 335-391 | DOI | MR | Zbl

[50] Fogg, N. Pytheas Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer, 2002, xviii+402 pages | MR | Zbl

[51] Frougny, Christiane; Solomyak, Boris Finite beta-expansions, Ergodic Theory Dyn. Syst., Volume 12 (1992) no. 4, pp. 713-723 | DOI | MR | Zbl

[52] Fujita, Takahiko; Ito, Shunji; Keane, Michael; Ohtsuki, Makoto On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof, Ergodic Theory Dyn. Syst., Volume 16 (1996) no. 6, pp. 1345-1352 | DOI | MR | Zbl

[53] Furstenberg, Harry Stationary processes and prediction theory, Annals of Mathematics Studies, 44, Princeton University Press, 1960, x+283 pages | MR | Zbl

[54] Furstenberg, Harry; Keynes, Harvey; Shapiro, Leonard Prime flows in topological dynamics, Isr. J. Math., Volume 14 (1973), pp. 26-38 | DOI | MR | Zbl

[55] Gorodnik, Alexander Open problems in dynamics and related fields, J. Mod. Dyn., Volume 1 (2007) no. 1, pp. 1-35 | MR | Zbl

[56] Grepstad, Sigrid; Lev, Nir Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal., Volume 25 (2015), pp. 87-133 | DOI | MR | Zbl

[57] Host, Bernard Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Ergodic Theory Dyn. Syst., Volume 6 (1986) no. 4, pp. 529-540 | DOI | Zbl

[58] Hubert, Pascal; Messaoudi, Ali Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals, Acta Arith., Volume 124 (2006) no. 1, pp. 1-15 | DOI | MR | Zbl

[59] Ito, Shunji Weyl automorphisms, substitutions and fractals, Stability theory and related topics in dynamical systems (Nagoya, 1988) (World Scientific Advanced Series in Dynamical Systems), Volume 6, World Scientific, 1989, pp. 60-72 | MR | Zbl

[60] Ito, Shunji Fractal domains of quasi-periodic motions on $T^2$, Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), Plenum Press, 1995, pp. 95-99 | Zbl

[61] Ito, Shunji; Fujii, Junko; Higashino, Hiroko; Yasutomi, Shin-Ichi On simultaneous approximation to $(\alpha ,{\alpha }^2)$ with ${\alpha }^3+k\alpha -1=0$, J. Number Theory, Volume 99 (2003) no. 2, pp. 255-283 | Zbl

[62] Ito, Shunji; Ohtsuki, Makoto Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math., Volume 16 (1993) no. 2, pp. 441-472 | MR | Zbl

[63] Ito, Shunji; Ohtsuki, Makoto Parallelogram tilings and Jacobi-Perron algorithm, Tokyo J. Math., Volume 17 (1994) no. 1, pp. 33-58 | MR | Zbl

[64] Ito, Shunji; Rao, Hui Atomic surfaces, tilings and coincidence. I. Irreducible case, Isr. J. Math., Volume 153 (2006), pp. 129-155 | MR | Zbl

[65] Ito, Shunji; Yasutomi, Shin-Ichi On simultaneous Diophantine approximation to periodic points related to modified Jacobi-Perron algorithm, Probability and number theory—Kanazawa 2005 (Advanced Studies in Pure Mathematics), Volume 49, Mathematical Society of Japan, 2007, pp. 171-184 | MR | Zbl

[66] Labbé, Sébastien; Leroy, Julien Bispecial factors in the Brun $S$-adic system, Developments in Language Theory (DLT) (Lecture Notes in Computer Science), Springer, 2016 | DOI | Zbl

[67] Lagarias, Jeffrey C. The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms, Monatsh. Math., Volume 115 (1993) no. 4, pp. 299-328 | DOI | MR | Zbl

[68] Meester, Ronald A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 4, pp. 1077-1083 | DOI | MR | Zbl

[69] Minervino, Milton; Thuswaldner, Jörg M. The geometry of non-unit Pisot substitutions, Ann. Inst. Fourier, Volume 64 (2014), pp. 1373-1417 | DOI | MR | Zbl

[70] Perron, Oskar Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann., Volume 64 (1907) no. 1, pp. 1-76 | MR | Zbl

[71] Podsypanin, E. V. A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Volume 67 (1977), pp. 184-194 | MR | Zbl

[72] Priebe Frank, Natalie; Sadun, Lorenzo Fusion: a general framework for hierarchical tilings of ${ℝ}^d$, Geom. Dedicata, Volume 171 (2014), pp. 149-186 | MR | Zbl

[73] Priebe Frank, Natalie; Sadun, Lorenzo Fusion tilings with infinite local complexity, Topol. Proc., Volume 43 (2014), pp. 235-276 | MR | Zbl

[74] Queffélec, Martine Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, 1294, Springer, 2010, xvi+351 pages | MR | Zbl

[75] Rauzy, Gérard Nombres algébriques et substitutions, Bull. Soc. Math. Fr., Volume 110 (1982) no. 2, pp. 147-178 | DOI | Zbl

[76] Rauzy, Gérard Ensembles à restes bornés, Seminar on number theory, 1983–1984 (Talence, 1983/1984), Université Bordeaux I, 1984 (Exp. No. 24, 12 pages) | Zbl

[77] Reveillès, Jean-Pierre Géométrie discrète, calculs en nombres entiers et algorithmes, Ph. D. Thesis, Université Louis Pasteur, Strasbourg (France) (1991) | Zbl

[78] Risley, Rebecca N.; Zamboni, Luca Q. A generalization of Sturmian sequences: combinatorial structure and transcendence, Acta Arith., Volume 95 (2000) no. 2, pp. 167-184 | DOI | MR | Zbl

[79] Sadun, Lorenzo Finitely balanced sequences and plasticity of $1$-dimensional Tilings, Topology Appl., Volume 205 (2016), pp. 82-87 | DOI | MR | Zbl

[80] Schratzberger, Bernhard R. The exponent of convergence for Brun’s algorithm in two dimensions, Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl., Volume 207 (1998), pp. 229-238 | MR | Zbl

[81] Schweiger, Fritz Invariant measures for maps of continued fraction type, J. Number Theory, Volume 39 (1991) no. 2, pp. 162-174 | MR | Zbl

[82] Schweiger, Fritz Multidimensional continued fractions, Oxford Science Publications, Oxford University Press, 2000, viii+234 pages | Zbl

[83] Sidorov, Nikita Arithmetic dynamics, Topics in dynamics and ergodic theory (London Mathematical Society Lecture Note Series), Volume 310, Cambridge University Press, 2003, pp. 145-189 | DOI | MR | Zbl

[84] Siegel, Anne; Thuswaldner, Jörg M. Topological properties of Rauzy fractals, Mém. Soc. Math. Fr., Nouv. Sér. (2009) no. 118 (140 pages) | MR | Zbl

[85] Sirvent, Víctor F.; Wang, Yang Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pac. J. Math., Volume 206 (2002) no. 2, pp. 465-485 | DOI | MR | Zbl

[86] Solomyak, Boris Dynamics of self-similar tilings, Ergodic Theory Dyn. Syst., Volume 17 (1997) no. 3, pp. 695-738 | DOI | MR | Zbl

[87] Vershik, Anatoliĭ M. Uniform algebraic approximation of shift and multiplication operators, Dokl. Akad. Nauk SSSR, Volume 259 (1981) no. 3, pp. 526-529 English translation in Sov. Math. Dokl. 24 (1981), p. 97–100 | MR | Zbl