Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions

Pierre Arnoux[1]; Valérie Berthé[1]; Shunji Ito[2]

  • [1] Institut de Mathématiques de Luminy, Campus de Luminy, Case 901, 13288 Marseille Cedex 9 (France)
  • [2] Tsuda College, Tsuda Machi, Kodaira, Tokyo 187 (Japon)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 305-349
  • ISSN: 0373-0956

Abstract

top
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 ℤ 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 ℤ 2 -actions by rotations.

How to cite

top

Arnoux, Pierre, Berthé, Valérie, and Ito, Shunji. "Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions." Annales de l’institut Fourier 52.2 (2002): 305-349. <http://eudml.org/doc/115982>.

@article{Arnoux2002,
abstract = {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 $\{\mathbb \{Z\}\}^2$-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 $\{\mathbb \{Z\}\}^\{2\}$-actions by rotations.},
affiliation = {Institut de Mathématiques de Luminy, Campus de Luminy, Case 901, 13288 Marseille Cedex 9 (France); Institut de Mathématiques de Luminy, Campus de Luminy, Case 901, 13288 Marseille Cedex 9 (France); Tsuda College, Tsuda Machi, Kodaira, Tokyo 187 (Japon)},
author = {Arnoux, Pierre, Berthé, Valérie, Ito, Shunji},
journal = {Annales de l’institut Fourier},
keywords = {substitutions; generalized continued fractions; discrete plans; tilings; Jacobi-Perron algorithm; induction; $\{\mathbb \{Z\}\}^2$-actions; two-dimensional sequences; discrete planes; -actions},
language = {eng},
number = {2},
pages = {305-349},
publisher = {Association des Annales de l'Institut Fourier},
title = {Discrete planes, $\{\mathbb \{Z\}\}^2$-actions, Jacobi-Perron algorithm and substitutions},
url = {http://eudml.org/doc/115982},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Arnoux, Pierre
AU - Berthé, Valérie
AU - Ito, Shunji
TI - Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 305
EP - 349
AB - 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 ${\mathbb {Z}}^2$-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 ${\mathbb {Z}}^{2}$-actions by rotations.
LA - eng
KW - substitutions; generalized continued fractions; discrete plans; tilings; Jacobi-Perron algorithm; induction; ${\mathbb {Z}}^2$-actions; two-dimensional sequences; discrete planes; -actions
UR - http://eudml.org/doc/115982
ER -

References

top
  1. P. Arnoux, Chaos from order, a worked out example, Complex Systems (2001), 1-67, Kluwer Academic Publ. Zbl1333.37006
  2. P. Arnoux, Sturmian sequences, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag Zbl1333.37006
  3. P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181-207 Zbl1007.37001MR1838930
  4. P. Arnoux, S. Ferenczi, P. Hubert, Trajectories of rotations, Acta Arith. 87 (1999), 209-217 Zbl0921.11033MR1668554
  5. P. Arnoux, S. Ito, Y. Sano, Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math. 83 (2001), 183-206 Zbl0987.11013MR1828491
  6. P. Arnoux, G. Rauzy, Représentation géométrique de suites de complexité 2 n + 1 , Bull. Soc. Math. France 119 (1991), 199-215 Zbl0789.28011MR1116845
  7. J. Berstel, Tracé de droites, fractions continues et morphismes itérés, Mots, Lang. Raison. Calc. (1990), 298-309, Éditions Hermès, Paris 
  8. J. Berstel, Recent results in Sturmian words, Developments in Language Theory II (1996), 13-24, World Scientific Zbl1096.68689
  9. J. Berstel, P. Séébold, Chapter 2: Sturmian words in M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press 
  10. V. Berthé, L. Vuillon, Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math. 223 (2000), 27-53 Zbl0970.68124MR1782038
  11. V. Berthé, L. Vuillon, Suites doubles de basse complexité, J. Th. Nombres Bordeaux 12 (2000), 179-208 Zbl1018.37010MR1827847
  12. V. Berthé, L. Vuillon, Palindromes and two-dimensional Sturmian sequences, J. Auto. Lang. Comp. 6 (2001), 121-138 Zbl1002.11026MR1828855
  13. A.J. Brentjes, Multi-dimensional continued fraction algorithms, 145 (1981), Matematisch Centrum, Amsterdam Zbl0471.10024
  14. A. Broise, Fractions continues multidimensionnelles et lois stables, Bull. Soc. Math. France 124 (1999), 97-139 Zbl0857.11035MR1395008
  15. A. Broise-Alamichei, Y. Guivarc'h, Exposants caratéristiques de l'algorithme de Jacobi-Perron et la transformation associée, Ann. Inst. Fourier 51 (2001), 565-686 Zbl1012.11060MR1838461
  16. T.C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull. 36 (1993), 15-21 Zbl0804.11021MR1205889
  17. V. Canterini, A. Siegel, Geometric representations of primitive substitutions of Pisot type Zbl1142.37302MR1852097
  18. J.H. Conway, C. Radin, Quaquaversal tilings and rotations, Inventiones Math. 132 (1998), 179-188 Zbl0913.52009MR1618635
  19. F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), 89-101 Zbl0895.68087MR1489074
  20. J. Françon, Sur la topologie d'un plan arithmétique, Th. Comput. Sci. 156 (1996), 159-176 Zbl0871.68165MR1382845
  21. D. Giammarresi, A. Restivo, Two-dimensional Languages, Handbook of Formal languages vol. 3 (1997), Springer-Verlag, Berlin Zbl0843.68054
  22. C. Goodman-Strauss, Matching rules and substitution tilings, Annals of Math. 147 (1998), 181-223 Zbl0941.52018MR1609510
  23. S. Ito, M. Kimura, On Rauzy fractal, Japan J. Indust. Appl. Math. 8 (1991), 461-486 Zbl0734.28010MR1137652
  24. S. Ito, M. Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math. 16 (1993), 441-472 Zbl0805.11056MR1247666
  25. S. Ito, M. Ohtsuki, Parallelogram tilings and Jacobi-Perron algorithm, Tokyo J. Math. 17 (1994), 33-58 Zbl0805.52011MR1279568
  26. A.B. Katok, A.M. Stepin, Approximations in ergodic theory, Usp. Math. Nauk. (in Russian) 22 (1967), 81-106 Zbl0172.07202MR219697
  27. A. Messaoudi, Frontière du fractal de Rauzy et système de numération complexe, Acta Arith. 95 (2000), 195-224 Zbl0968.28005MR1793161
  28. M. Morse, G.A. Hedlund, Symbolic dynamics II: Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42 Zbl0022.34003MR745
  29. N. Priebe, Towards a characterization of self-similar tilings in terms of derived Voronoï tessellations, Geom. Dedicata 79 (2000), 239-265 Zbl1048.37014MR1755727
  30. M. Queffélec, Substitution dynamical systems, Spectral analysis 1294 (1987), Springer-Verlag Zbl0642.28013
  31. C. Radin, Space tilings and substitutions, Geom. Dedicata 55 (1995), 257-264 Zbl0835.52018MR1334449
  32. C. Radin, Miles of tiles, Vol. 1 (1999), Amer. Math. Soc., Providence Zbl0932.52005MR1707270
  33. C. Radin, A homeomorphism invariant for substitution tiling spaces Zbl0997.37006MR1898159
  34. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178 Zbl0522.10032MR667748
  35. J.-P. Reveillès, Combinatorial pieces in digital lines and planes, Vision geometry IV (San Diego, CA, 1995) 2573, 23-24 
  36. O. Salon, Suites automatiques à multi-indices, Sém. Th. Nombres Bordeaux exp. no 4 (1986-1987), Univ. Bordeaux I Zbl0653.10049
  37. O. Salon, Suites automatiques à multi-indices et algébricité, C. R. Acad. Sci. Paris, Sér. I Math. 305 (1987), 501-504 Zbl0628.10007MR916320
  38. M. Senechal, Quasicrystals and geometry, (1995), Cambridge University Press Zbl0828.52007MR1340198
  39. F. Schweiger, The metrical theory of Jacobi-Perron algorithm, 334 (1973), Springer-Verlag Zbl0287.10041MR345925
  40. J.-L. Verger-Gaugry, J.-P. Gazeau, Geometric study of the set ℤ β of beta-integers with β a Perron number, a β -number and a Pisot number and mathematical quasicrystals, (2000) 
  41. L. Vuillon, Combinatoire des motifs d'une suite sturmienne bidimensionnelle, Th. Comput. Sci. 209 (1998), 261-285 Zbl0913.68206MR1647534
  42. A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres Bordeaux 10 (1998), 135-162 Zbl0918.11048MR1827290
  43. A.B. Katok, A.M. Stepin, Approximations in ergodic theory, Russian Math. Surveys 22 (1967), 76-102 Zbl0172.07202MR219697

Citations in EuDML Documents

top
  1. N. Pytheas Fogg, Substitutions par des motifs en dimension 1
  2. Valérie Berthé, Discrete geometry and numeration
  3. Hiromi Ei, Shunji Ito, Hui Rao, Atomic surfaces, tilings and coincidences II. Reducible case
  4. Valérie Berthé, Timo Jolivet, Anne Siegel, Connectedness of fractals associated with Arnoux–Rauzy substitutions
  5. Julien Bernat, Study of irreducible balanced pairs for substitutive languages
  6. Pierre Arnoux, Valérie Berthé, Arnaud Hilion, Anne Siegel, Fractal representation of the attractive lamination of an automorphism of the free group

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.