Geometry, dynamics, and arithmetic of S-adic shifts
[Géométrie, dynamique, et arithmétique des décalages S-adiques]
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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3273
Classification : 37B10,  37A30,  11K50,  28A80
Mots 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
@article{AIF_2019__69_3_1347_0,
     author = {Berth\'e, Val\'erie and Steiner, Wolfgang and Thuswaldner, J\"org M.},
     title = {Geometry, dynamics, and arithmetic of $S$-adic shifts},
     journal = {Annales de l'Institut Fourier},
     pages = {1347--1409},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     doi = {10.5802/aif.3273},
     zbl = {07067434},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3273/}
}
Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jörg M. Geometry, dynamics, and arithmetic of $S$-adic shifts. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1347-1409. doi : 10.5802/aif.3273. http://archive.numdam.org/articles/10.5802/aif.3273/

[1] Adamczewski, Boris Balances for fixed points of primitive substitutions, Theor. Comput. Sci., Volume 307 (2003) no. 1, pp. 47-75 | MR 2014730 | Zbl 1059.68083

[2] Adamczewski, Boris Symbolic discrepancy and self-similar dynamics, Ann. Inst. Fourier, Volume 54 (2004) no. 7, pp. 2201-2234 | Article | MR 2139693 | Zbl 1066.11032

[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 | Article | MR 3381478

[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 | Article | MR 2764877 | Zbl 1219.37013

[5] Arnoux, Pierre; Berthé, Valérie; Ito, Shunji Discrete planes, 2 -actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier, Volume 52 (2002) no. 2, pp. 305-349 | Article | MR 1906478 | Zbl 1017.11006

[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 | Article | MR 1870070 | Zbl 0993.37018

[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 | Article | MR 2142941 | Zbl 1140.37314

[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 1838930 | Zbl 1007.37001

[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 | Article | MR 3225711 | Zbl 06313777

[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 | Article | MR 1251147 | Zbl 0801.11036

[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 | Article | Zbl 0789.28011

[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 | Article | MR 3556526 | Zbl 1376.37030

[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 | Article | MR 3558432 | Zbl 1356.37018

[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 | Article | MR 3431249 | Zbl 1338.37014

[17] Barge, Marcy The Pisot conjecture for β-substitutions, Ergodic Theory Dyn. Syst., Volume 38 (2018), pp. 444-472 | Article | MR 3774828 | Zbl 06872477

[18] Barge, Marcy; Kwapisz, Jaroslaw Geometric theory of unimodular Pisot substitutions, Am. J. Math., Volume 128 (2006) no. 5, pp. 1219-1282 | Article | MR 2262174 | Zbl 1152.37011

[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 2975125 | Zbl 1291.37024

[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 | Article | MR 2681739 | Zbl 1149.68065

[21] Berthé, Valérie Multidimensional Euclidean algorithms, numeration and substitutions, Integers, Volume 11B (2011), A02, 34 pages (Art. ID A02, 34 pages) | MR 3054421 | Zbl 1273.11014

[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 (2013), pp. 58-70 | MR 3124511 | Zbl 1400.11075

[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 1378.37036

[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 | Article | MR 3078051 | Zbl 1269.68073

[25] Berthé, Valérie; Delecroix, Vincent Beyond substitutive dynamical systems: S-adic expansions, RIMS Kôkyûroku Bessatsu, Volume B46 (2014), pp. 81-123 | Zbl 1376.37033

[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 | Article | MR 2180244 | Zbl 1156.37301

[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 2943167 | Zbl 1313.37004

[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 | Article | MR 3493306 | Zbl 1368.37020

[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 | Article | MR 2759108 | Zbl 1247.37015

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

[31] Berthé, Valérie; Tijdeman, Robert Balance properties of multi-dimensional words, Theor. Comput. Sci., Volume 273 (2002) no. 1-2, pp. 197-224 | Article | MR 1872450 | Zbl 0997.68091

[32] Birkhoff, Garrett Extensions of Jentzsch’s theorem, Trans. Am. Math. Soc., Volume 85 (1957), pp. 219-227 | MR 87058 | Zbl 0079.13502

[33] Brentjes, Arne J. Multidimensional continued fraction algorithms, Mathematical Centre Tracts, 145, Mathematisch Centrum, 1981, i+183 pages | MR 638474 | Zbl 0471.10024

[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 2808407 | Zbl 1303.11077

[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 111735 | Zbl 0086.03204

[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 | Article | MR 2473626 | Zbl 1151.37013

[37] Cassaigne, Julien; Ferenczi, Sébastien; Zamboni, Luca Q. Imbalances in Arnoux–Rauzy sequences, Ann. Inst. Fourier, Volume 50 (2000) no. 4, pp. 1265-1276 | Article | MR 1799745 | Zbl 1004

[38] Chevallier, Nicolas Coding of a translation of the two-dimensional torus, Monatsh. Math., Volume 157 (2009) no. 2, pp. 101-130 | Article | MR 2504781 | Zbl 1171.37009

[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 | Article | MR 1997967 | Zbl 1042.37008

[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 | Article | MR 461470 | Zbl 0348.54034

[41] Delecroix, Vincent; Hejda, Tomáš; Steiner, Wolfgang Balancedness of Arnoux-Rauzy and Brun Words, WORDS (Lecture Notes in Computer Science), Volume 8079 (2013), pp. 119-131 | MR 3124516 | Zbl 1398.68414

[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 | Article | MR 3297155 | Zbl 1351.37159

[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 | Article | MR 1779393 | Zbl 0965.37013

[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 1972245

[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 | Article | MR 1709427 | Zbl 1044.46543

[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 3032389 | Zbl 1354.68214

[47] Ferenczi, Sébastien Bounded remainder sets, Acta Arith., Volume 61 (1992) no. 4, pp. 319-326 | Article | MR 1168091 | Zbl 0774.11037

[48] Fernique, Thomas Multidimensional Sturmian sequences and generalized substitutions, Int. J. Found. Comput. Sci., Volume 17 (2006) no. 3, pp. 575-600 | Article | MR 2234803 | Zbl 1096.68125

[49] Fisher, Albert M. Nonstationary mixing and the unique ergodicity of adic transformations, Stoch. Dyn., Volume 9 (2009) no. 3, pp. 335-391 | Article | MR 2566907 | Zbl 1231.28014

[50] Fogg, N. Pytheas Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer, 2002, xviii+402 pages | MR 1970385 | Zbl 1014.11015

[51] Frougny, Christiane; Solomyak, Boris Finite beta-expansions, Ergodic Theory Dyn. Syst., Volume 12 (1992) no. 4, pp. 713-723 | Article | MR 1200339 | Zbl 0814.68065

[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 | Article | MR 1424403 | Zbl 0868.28008

[53] Furstenberg, Harry Stationary processes and prediction theory, Annals of Mathematics Studies, 44, Princeton University Press, 1960, x+283 pages | MR 140151 | Zbl 0178.53002

[54] Furstenberg, Harry; Keynes, Harvey; Shapiro, Leonard Prime flows in topological dynamics, Isr. J. Math., Volume 14 (1973), pp. 26-38 | Article | MR 321055 | Zbl 0264.54030

[55] Gorodnik, Alexander Open problems in dynamics and related fields, J. Mod. Dyn., Volume 1 (2007) no. 1, pp. 1-35 | MR 2261070 | Zbl 1118.37002

[56] Grepstad, Sigrid; Lev, Nir Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal., Volume 25 (2015), pp. 87-133 | Article | MR 3320890 | Zbl 1318.11097

[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 | Article | Zbl 0625.28011

[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 | Article | MR 2262136 | Zbl 1116.28009

[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 1116818 | Zbl 0702.11047

[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 0865.28010

[61] Ito, Shunji; Fujii, Junko; Higashino, Hiroko; Yasutomi, Shin-Ichi On simultaneous approximation to (α,α 2 ) with α 3 +kα-1=0, J. Number Theory, Volume 99 (2003) no. 2, pp. 255-283 | Zbl 1135.11326

[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 1247666 | Zbl 0805.11056

[63] Ito, Shunji; Ohtsuki, Makoto Parallelogram tilings and Jacobi-Perron algorithm, Tokyo J. Math., Volume 17 (1994) no. 1, pp. 33-58 | MR 1279568 | Zbl 0805.52011

[64] Ito, Shunji; Rao, Hui Atomic surfaces, tilings and coincidence. I. Irreducible case, Isr. J. Math., Volume 153 (2006), pp. 129-155 | MR 2254640 | Zbl 1143.37013

[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 2405603 | Zbl 1223.11084

[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 | Article | Zbl 06620603

[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 | Article | MR 1230366 | Zbl 0790.11059

[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 | Article | MR 1709431 | Zbl 1044.11074

[69] Minervino, Milton; Thuswaldner, Jörg M. The geometry of non-unit Pisot substitutions, Ann. Inst. Fourier, Volume 64 (2014), pp. 1373-1417 | Article | MR 3329667 | Zbl 1309.05045

[70] Perron, Oskar Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann., Volume 64 (1907) no. 1, pp. 1-76 | MR 1511422 | Zbl 38.0262.01

[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 457337 | Zbl 0357.10016

[72] Priebe Frank, Natalie; Sadun, Lorenzo Fusion: a general framework for hierarchical tilings of d , Geom. Dedicata, Volume 171 (2014), pp. 149-186 | MR 3226791 | Zbl 1305.37010

[73] Priebe Frank, Natalie; Sadun, Lorenzo Fusion tilings with infinite local complexity, Topol. Proc., Volume 43 (2014), pp. 235-276 | MR 3104922 | Zbl 1329.37017

[74] Queffélec, Martine Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, 1294, Springer, 2010, xvi+351 pages | MR 2590264 | Zbl 1225.11001

[75] Rauzy, Gérard Nombres algébriques et substitutions, Bull. Soc. Math. Fr., Volume 110 (1982) no. 2, pp. 147-178 | Article | Zbl 0522.10032

[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 0547.10044

[77] Reveillès, Jean-Pierre Géométrie discrète, calculs en nombres entiers et algorithmes (1991) (Ph. D. Thesis) | Zbl 1079.51513

[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 | Article | MR 1785413 | Zbl 0953.11007

[79] Sadun, Lorenzo Finitely balanced sequences and plasticity of 1-dimensional Tilings, Topology Appl., Volume 205 (2016), pp. 82-87 | Article | MR 3493308 | Zbl 1356.37024

[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 1749922 | Zbl 1040.11510

[81] Schweiger, Fritz Invariant measures for maps of continued fraction type, J. Number Theory, Volume 39 (1991) no. 2, pp. 162-174 | MR 1129566 | Zbl 0748.11037

[82] Schweiger, Fritz Multidimensional continued fractions, Oxford Science Publications, Oxford University Press, 2000, viii+234 pages | Zbl 0981.11029

[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 | Article | MR 2052279 | Zbl 1051.37007

[84] Siegel, Anne; Thuswaldner, Jörg M. Topological properties of Rauzy fractals, Mém. Soc. Math. Fr., Nouv. Sér. (2009) no. 118 (140 pages) | MR 2721985 | Zbl 1229.28021

[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 | Article | MR 1926787 | Zbl 1048.37015

[86] Solomyak, Boris Dynamics of self-similar tilings, Ergodic Theory Dyn. Syst., Volume 17 (1997) no. 3, pp. 695-738 | Article | MR 1452190 | Zbl 0884.58062

[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 625756 | Zbl 0484.47005