Discrete planes, 2 -actions, Jacobi-Perron algorithm and substitutions
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, p. 305-349

We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a 2 -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of 2 -actions by rotations.

Nous définissons des substitutions bi-dimensionnelles; ces substitutions engendrent des suites doubles reliées à des approximations discrètes de plans irrationnels. Elles sont obtenues au moyen de l’algorithme classique de Jacobi Perron, en définissant l’induction d’une action de 2 par rotations sur le cercle. On donne ainsi une interprétation géométrique nouvelle de l’algorithme de Jacobi-Perron, comme application opérant sur l’espace des paramètres des actions de 2 par rotations.

DOI : https://doi.org/10.5802/aif.1889
Classification:  11A55,  11J70,  40A15,  68R15
Keywords: substitutions, generalized continued fractions, discrete plans, tilings, Jacobi-Perron algorithm, induction, 2 -actions, two-dimensional sequences
@article{AIF_2002__52_2_305_0,
     author = {Arnoux, Pierre and Berth\'e, Val\'erie and Ito, Shunji},
     title = {Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     pages = {305-349},
     doi = {10.5802/aif.1889},
     zbl = {1017.11006},
     mrnumber = {1906478},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_2_305_0}
}
Arnoux, Pierre; Berthé, Valérie; Ito, Shunji. Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 305-349. doi : 10.5802/aif.1889. http://www.numdam.org/item/AIF_2002__52_2_305_0/

[1] P. Arnoux; E. Goles And S. Martinez (Eds.) Chaos from order, a worked out example, Complex Systems, Kluwer Academic Publ. (2001), pp. 1-67

[2] P. Arnoux; V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel (Eds.) Sturmian sequences, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag (To appear in Lecture Notes in Math.)

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

[4] P. Arnoux; S. Ferenczi; P. Hubert Trajectories of rotations, Acta Arith., Tome 87 (1999), pp. 209-217 | MR 1668554 | Zbl 0921.11033

[5] P. Arnoux; S. Ito; Y. Sano Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math., Tome 83 (2001), pp. 183-206 | Article | MR 1828491 | Zbl 0987.11013

[6] P. Arnoux; G. Rauzy Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France, Tome 119 (1991), pp. 199-215 | Numdam | MR 1116845 | Zbl 0789.28011

[7] J. Berstel Tracé de droites, fractions continues et morphismes itérés, Mots, Lang. Raison. Calc., Éditions Hermès, Paris (1990), pp. 298-309

[8] J. Berstel; Dassow, Rozenberg, Salomaa, Eds. Recent results in Sturmian words, Developments in Language Theory II, World Scientific (1996), pp. 13-24 | Zbl 1096.68689

[9] J. Berstel; P. Séébold Chapter 2: Sturmian words in M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press (To appear)

[10] V. Berthé; L. Vuillon Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math., Tome 223 (2000), pp. 27-53 | Article | MR 1782038 | Zbl 0970.68124

[11] V. Berthé; L. Vuillon Suites doubles de basse complexité, J. Th. Nombres Bordeaux, Tome 12 (2000), pp. 179-208 | Article | Numdam | MR 1827847 | Zbl 1018.37010

[12] V. Berthé; L. Vuillon Palindromes and two-dimensional Sturmian sequences, J. Auto. Lang. Comp., Tome 6 (2001), pp. 121-138 | MR 1828855 | Zbl 1002.11026

[13] A.J. Brentjes Multi-dimensional continued fraction algorithms, Matematisch Centrum, Amsterdam, Mathematical Centre Tracts, Tome 145 (1981) | Zbl 0471.10024

[14] A. Broise Fractions continues multidimensionnelles et lois stables, Bull. Soc. Math. France, Tome 124 (1999), pp. 97-139 | Numdam | MR 1395008 | Zbl 0857.11035

[15] A. Broise-Alamichei; Y. Guivarc'H Exposants caratéristiques de l'algorithme de Jacobi-Perron et la transformation associée, Ann. Inst. Fourier, Tome 51 (2001) no. 3, pp. 565-686 | Article | Numdam | MR 1838461 | Zbl 1012.11060

[16] T.C. Brown Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull., Tome 36 (1993), pp. 15-21 | Article | MR 1205889 | Zbl 0804.11021

[17] V. Canterini; A. Siegel Geometric representations of primitive substitutions of Pisot type (To appear in Trans. Amer. Math. Soc.) | MR 1852097 | Zbl 01663181

[18] J.H. Conway; C. Radin Quaquaversal tilings and rotations, Inventiones Math., Tome 132 (1998), pp. 179-188 | Article | MR 1618635 | Zbl 0913.52009

[19] F. Durand A characterization of substitutive sequences using return words, Discrete Math., Tome 179 (1998), pp. 89-101 | Article | MR 1489074 | Zbl 0895.68087

[20] J. Françon Sur la topologie d'un plan arithmétique, Th. Comput. Sci., Tome 156 (1996), pp. 159-176 | Article | MR 1382845 | Zbl 0871.68165

[21] D. Giammarresi; A. Restivo; A. Salomaa, G. Rozenberg, Eds. Two-dimensional Languages, Handbook of Formal languages, Springer-Verlag, Berlin, Tome vol. 3 (1997)

[22] C. Goodman-Strauss Matching rules and substitution tilings, Annals of Math., Tome 147 (1998), pp. 181-223 | Article | MR 1609510 | Zbl 0941.52018

[23] S. Ito; M. Kimura On Rauzy fractal, Japan J. Indust. Appl. Math., Tome 8 (1991), pp. 461-486 | Article | MR 1137652 | Zbl 0734.28010

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

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

[26] A.B. Katok; A.M. Stepin Approximations in ergodic theory, Usp. Math. Nauk. (in Russian), Tome 22 (1967), pp. 81-106 | MR 219697 | Zbl 0172.07202

[26] A.B. Katok; A.M. Stepin Approximations in ergodic theory, Russian Math. Surveys, Tome 22 (1967), pp. 76-102 | MR 219697 | Zbl 0172.07202

[27] A. Messaoudi Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres Bordeaux, Tome 10 (1998), pp. 135-162 | Article | Numdam | MR 1827290 | Zbl 0918.11048

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

[29] M. Morse; G.A. Hedlund Symbolic dynamics II: Sturmian trajectories, Amer. J. Math., Tome 62 (1940), pp. 1-42 | Article | JFM 66.0188.03 | MR 745 | Zbl 0022.34003

[30] N. Priebe Towards a characterization of self-similar tilings in terms of derived Voronoï tessellations, Geom. Dedicata, Tome 79 (2000), pp. 239-265 | Article | MR 1755727 | Zbl 1048.37014

[31] M. Queffélec Substitution dynamical systems, Spectral analysis, Springer-Verlag (Lecture Notes in Math.) Tome 1294 (1987) | Zbl 0642.28013

[32] C. Radin Space tilings and substitutions, Geom. Dedicata, Tome 55 (1995), pp. 257-264 | Article | MR 1334449 | Zbl 0835.52018

[33] C. Radin Miles of tiles, Amer. Math. Soc., Providence, Student Mathematical Library, Tome Vol. 1 (1999) | MR 1707270 | Zbl 0932.52005

[34] C. Radin A homeomorphism invariant for substitution tiling spaces (To appear in Geom. Dedicata) | MR 1898159 | Zbl 0997.37006

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

[36] J.-P. Reveillès Combinatorial pieces in digital lines and planes, Vision geometry IV (San Diego, CA, 1995) (Proc. SPIE) Tome 2573, p. 23-24

[37] O. Salon Suites automatiques à multi-indices, Sém. Th. Nombres Bordeaux, Univ. Bordeaux I, Tome exp. no 4 (1986-1987) | Zbl 0653.10049

[38] O. Salon Suites automatiques à multi-indices et algébricité, C. R. Acad. Sci. Paris, Sér. I Math., Tome 305 (1987), pp. 501-504 | MR 916320 | Zbl 0628.10007

[39] M. Senechal Quasicrystals and geometry, Cambridge University Press (1995) | MR 1340198 | Zbl 0828.52007

[40] F. Schweiger The metrical theory of Jacobi-Perron algorithm, Springer-Verlag, Lecture Notes in Math., Tome 334 (1973) | MR 345925 | Zbl 0287.10041

[41] J.-L. Verger-Gaugry; J.-P. Gazeau Geometric study of the set β of beta-integers with β a Perron number, a β-number and a Pisot number and mathematical quasicrystals (2000) (Prépublication de l'Institut Fourier, 513)

[42] L. Vuillon Combinatoire des motifs d'une suite sturmienne bidimensionnelle, Th. Comput. Sci., Tome 209 (1998), pp. 261-285 | Article | MR 1647534 | Zbl 0913.68206