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 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 par rotations.
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 -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 -actions by rotations.
Classification : 11A55, 11J70, 40A15, 68R15
Mots clés : substitutions, fractions continues généralisées, plans discrets, pavages, algorithme de Jacobi-Perron, induction, actions de , suites doubles
@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}, pages = {305--349}, publisher = {Association des Annales de l'institut Fourier}, volume = {52}, number = {2}, year = {2002}, doi = {10.5802/aif.1889}, zbl = {1017.11006}, mrnumber = {1906478}, language = {en}, url = {archive.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, Tome 52 (2002) no. 2, pp. 305-349. doi : 10.5802/aif.1889. http://archive.numdam.org/item/AIF_2002__52_2_305_0/
[1] Chaos from order, a worked out example, Complex Systems (2001), pp. 1-67
[2] Sturmian sequences, Substitutions in Dynamics, Arithmetics and Combinatorics (To appear in Lecture Notes in Math.)
[3] Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Volume 8 (2001), pp. 181-207 | MR 1838930 | Zbl 1007.37001
[4] Trajectories of rotations, Acta Arith., Volume 87 (1999), pp. 209-217 | MR 1668554 | Zbl 0921.11033
[5] Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math., Volume 83 (2001), pp. 183-206 | Article | MR 1828491 | Zbl 0987.11013
[6] Représentation géométrique de suites de complexité , Bull. Soc. Math. France, Volume 119 (1991), pp. 199-215 | Numdam | MR 1116845 | Zbl 0789.28011
[7] Tracé de droites, fractions continues et morphismes itérés, Mots, Lang. Raison. Calc. (1990), pp. 298-309
[8] Recent results in Sturmian words, Developments in Language Theory II (1996), pp. 13-24 | Zbl 1096.68689
[9] Chapter 2: Sturmian words in M. Lothaire, Algebraic Combinatorics on Words (To appear)
[10] Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math., Volume 223 (2000), pp. 27-53 | Article | MR 1782038 | Zbl 0970.68124
[11] Suites doubles de basse complexité, J. Th. Nombres Bordeaux, Volume 12 (2000), pp. 179-208 | Article | Numdam | MR 1827847 | Zbl 1018.37010
[12] Palindromes and two-dimensional Sturmian sequences, J. Auto. Lang. Comp., Volume 6 (2001), pp. 121-138 | MR 1828855 | Zbl 1002.11026
[13] Multi-dimensional continued fraction algorithms, Mathematical Centre Tracts, Volume 145, Matematisch Centrum, Amsterdam, 1981 | Zbl 0471.10024
[14] Fractions continues multidimensionnelles et lois stables, Bull. Soc. Math. France, Volume 124 (1999), pp. 97-139 | Numdam | MR 1395008 | Zbl 0857.11035
[15] Exposants caratéristiques de l'algorithme de Jacobi-Perron et la transformation associée, Ann. Inst. Fourier, Volume 51 (2001) no. 3, pp. 565-686 | Article | Numdam | MR 1838461 | Zbl 1012.11060
[16] Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull., Volume 36 (1993), pp. 15-21 | Article | MR 1205889 | Zbl 0804.11021
[17] Geometric representations of primitive substitutions of Pisot type (To appear in Trans. Amer. Math. Soc.) | MR 1852097 | Zbl 01663181
[18] Quaquaversal tilings and rotations, Inventiones Math., Volume 132 (1998), pp. 179-188 | Article | MR 1618635 | Zbl 0913.52009
[19] A characterization of substitutive sequences using return words, Discrete Math., Volume 179 (1998), pp. 89-101 | Article | MR 1489074 | Zbl 0895.68087
[20] Sur la topologie d'un plan arithmétique, Th. Comput. Sci., Volume 156 (1996), pp. 159-176 | Article | MR 1382845 | Zbl 0871.68165
[21] Two-dimensional Languages, Handbook of Formal languages, Volume vol. 3 (1997)
[22] Matching rules and substitution tilings, Annals of Math., Volume 147 (1998), pp. 181-223 | Article | MR 1609510 | Zbl 0941.52018
[23] On Rauzy fractal, Japan J. Indust. Appl. Math., Volume 8 (1991), pp. 461-486 | Article | MR 1137652 | Zbl 0734.28010
[24] Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math., Volume 16 (1993), pp. 441-472 | Article | MR 1247666 | Zbl 0805.11056
[25] Parallelogram tilings and Jacobi-Perron algorithm, Tokyo J. Math., Volume 17 (1994), pp. 33-58 | Article | MR 1279568 | Zbl 0805.52011
[26] Approximations in ergodic theory, Usp. Math. Nauk. (in Russian), Volume 22 (1967), pp. 81-106 | MR 219697 | Zbl 0172.07202
[26] Approximations in ergodic theory, Russian Math. Surveys, Volume 22 (1967), pp. 76-102 | MR 219697 | Zbl 0172.07202
[27] Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres Bordeaux, Volume 10 (1998), pp. 135-162 | Article | Numdam | MR 1827290 | Zbl 0918.11048
[28] Frontière du fractal de Rauzy et système de numération complexe, Acta Arith., Volume 95 (2000), pp. 195-224 | MR 1793161 | Zbl 0968.28005
[29] Symbolic dynamics II: Sturmian trajectories, Amer. J. Math., Volume 62 (1940), pp. 1-42 | Article | JFM 66.0188.03 | MR 745 | Zbl 0022.34003
[30] Towards a characterization of self-similar tilings in terms of derived Voronoï tessellations, Geom. Dedicata, Volume 79 (2000), pp. 239-265 | Article | MR 1755727 | Zbl 1048.37014
[31] Substitution dynamical systems, Spectral analysis (Lecture Notes in Math.) Volume 1294 (1987) | Zbl 0642.28013
[32] Space tilings and substitutions, Geom. Dedicata, Volume 55 (1995), pp. 257-264 | Article | MR 1334449 | Zbl 0835.52018
[33] Miles of tiles, Student Mathematical Library, Volume Vol. 1, Amer. Math. Soc., Providence, 1999 | MR 1707270 | Zbl 0932.52005
[34] A homeomorphism invariant for substitution tiling spaces (To appear in Geom. Dedicata) | MR 1898159 | Zbl 0997.37006
[35] Nombres algébriques et substitutions, Bull. Soc. Math. France, Volume 110 (1982), pp. 147-178 | Numdam | MR 667748 | Zbl 0522.10032
[36] Combinatorial pieces in digital lines and planes, Vision geometry IV (San Diego, CA, 1995) (Proc. SPIE) Volume 2573, p. 23-24
[37] Suites automatiques à multi-indices, Sém. Th. Nombres Bordeaux, Volume exp. no 4 (1986-1987) | Zbl 0653.10049
[38] Suites automatiques à multi-indices et algébricité, C. R. Acad. Sci. Paris, Sér. I Math., Volume 305 (1987), pp. 501-504 | MR 916320 | Zbl 0628.10007
[39] Quasicrystals and geometry, Cambridge University Press, 1995 | MR 1340198 | Zbl 0828.52007
[40] The metrical theory of Jacobi-Perron algorithm, Lecture Notes in Math., Volume 334, Springer-Verlag, 1973 | MR 345925 | Zbl 0287.10041
[41] Geometric study of the set of beta-integers with a Perron number, a -number and a Pisot number and mathematical quasicrystals (2000) (Prépublication de l'Institut Fourier, 513)
[42] Combinatoire des motifs d'une suite sturmienne bidimensionnelle, Th. Comput. Sci., Volume 209 (1998), pp. 261-285 | Article | MR 1647534 | Zbl 0913.68206