Search and download archives of mathematical journals

  
 
  Table of contents for this issue | Previous article | Next article
Cassaigne, Julien; Ferenczi, Sébastien; Zamboni, Luca Q.
Imbalances in Arnoux-Rauzy sequences. Annales de l'institut Fourier, 50 no. 4 (2000), p. 1265-1276
Full text djvu | pdf | Reviews MR 2001j:68097 | Zbl 01478822 | 5 citations in Numdam

stable URL: http://www.numdam.org/item?id=AIF_2000__50_4_1265_0

Lookup this article on the publisher's site

Abstract

In a 1982 paper Rauzy showed that the subshift $(X,T)$ generated by the morphism $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$ is a natural coding of a rotation on the two-dimensional torus ${\Bbb T}^2$, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ${\Bbb R}^2,$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2n+1$ satisfying a combinatorial criterion known as the $\star $ condition of Arnoux and Rauzy codes the orbit of a point under a rotation on ${\Bbb T}^2$. In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence $\omega _* $ which is unbalanced in the following sense: for each $N>0$ there exist two factors of $\omega _* $ of equal length, with one having at least $N$ more occurrences of a given letter than the other. We then invoke a result due to Rauzy on bounded remainder sets to establish the existence of an Arnoux-Rauzy sequence which is not a natural coding of a rotation on ${\Bbb T}^2$.

Bibliography

[1] P. ARNOUX, Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore, Bull. Soc. Math. France, 116 (1988), 489-500.
Numdam |  MR 91a:58138 |  Zbl 0703.58045
[2] P. ARNOUX, V. BERHÉ, S. ITO, Discrete planes, ℤ2-actions, Jacobi-Perron algorithm and substitutions, preprint (1999).
[3] P. ARNOUX, S. ITO, Pisot substitutions and Rauzy fractals, preprint 98-18, Institut de Mathématiques de Luminy, Marseille (1998).
[4] P. ARNOUX, G. RAUZY, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France, 119 (1991), 199-215.
Numdam |  MR 92k:58072 |  Zbl 0789.28011
[5] V. BERTHÉ, L. VUILLON, Tilings and rotations: a two-dimensional generalization of Sturmian sequences, preprint 97-19, Institut de Mathématiques de Luminy, Marseille (1997).
[6] V. CANTERINI, A. SIEGEL, Geometric representations of Pisot substitutions, preprint (1999).  Zbl 01663181
[7] M.G. CASTELLI, F. MIGNOSI, A. RESTIVO, Fine and Wilf's theorem for three periods and a generalization of Sturmian words, Theoret. Comp. Sci., 218 (1999), 83-94.  MR 2000c:68110 |  Zbl 0916.68114
[8] R. V. CHACON, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562.  MR 40 #297 |  Zbl 0186.37203
[9] N. CHEKHOVA, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une question d'Hedlund et Morse, preprint (1999).
[10] N. CHEKHOVA, Algorithme d'approximation et propriétés ergodiques des suites d'Arnoux-Rauzy, preprint (1999).
[11] N. CHEKHOVA, P. HUBERT, A. MESSAOUDI, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, preprint 98-24, Institut de Mathématiques de Luminy, Marseille (1998).
[12] X. DROUBAY, J. JUSTIN, G. PIRILLO, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comp. Sci., to appear.  Zbl 0981.68126
[13] S. FERENCZI, Bounded remainder sets, Acta Arith., 61 (1992), 319-326.
Article |  MR 93f:11059 |  Zbl 0774.11037
[14] S. FERENCZI, Les transformations de Chacon : combinatoire, structure géométrique, lien avec les systèmes de complexité 2n + 1, Bull. Soc. Math. France, 123 (1995), 271-292.
Numdam |  MR 96m:28018 |  Zbl 0855.28008
[15] S. FERENCZI, C. HOLTON, L.Q. ZAMBONI, Structure of three interval exchange transformations I: An arithmetic study, preprint 199/2000, Université François Rabelais, Tours (2000).  Zbl 01653449
[16] S. FERENCZI, C. HOLTON, L.Q. ZAMBONI, Structure of three interval exchange transformations II: A combinatorial study; ergodic and spectral properties, preprint (2000).  Zbl 01653449
[17] S. FERENCZI, C. MAUDUIT, Transcendence of numbers with a low complexity expansion, J. Number Theory, 67 (1997), 146-161.  MR 98m:11079 |  Zbl 0895.11029
[18] S. ITO, M. KIMURA, On Rauzy fractal, Japan J. Indust. Appl. Math., 8 (1991), 461-486.  MR 93d:11084 |  Zbl 0734.28010
[19] H. KESTEN, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith., 12 (1966), 193-212.
Article |  MR 35 #155 |  Zbl 0144.28902
[20] M. LOTHAIRE, Algebraic Combinatorics on Words, Chapter 2: Sturmian words, by J. Berstel, P. Séébold, to appear.  Zbl 01737190
[21] A. MESSAOUDI, Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres de Bordeaux, 10 (1998), 135-162.
Numdam |  MR 2002c:11091 |  Zbl 0918.11048
[22] A. MESSAOUDI, Frontière du fractal de Rauzy et système de numération complexe, Acta Arith., to appear.
Article |  Zbl 0968.28005
[23] M. MORSE, G.A. HEDLUND, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866.  Zbl 0019.33502 |  JFM 64.0798.04
[24] M. MORSE, G.A. HEDLUND, Symbolic dynamics II: Sturmian sequences, Amer. J. Math., 62 (1940), 1-42.  MR 1,123d |  Zbl 0022.34003 |  JFM 66.0188.03
[25] G. RAUZY, Une généralisation du développement en fraction continue, Séminaire Delange - Pisot - Poitou, 1976-1977, Paris, exposé 15, 15-01-15-16.
Numdam |  MR 83e:10077 |  Zbl 0369.28015
[26] G. RAUZY, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
Article |  MR 82m:10076 |  Zbl 0414.28018
[27] G. RAUZY, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
Numdam |  MR 84h:10074 |  Zbl 0522.10032
[28] G. RAUZY, Ensembles à restes bornés, Séminaire de Théorie des Nombres de Bordeaux, 1983-1984, exposé 24, 24-01-24-12.
Article |  MR 86g:28024 |  Zbl 0547.10044
[29] R. RISLEY, L.Q. ZAMBONI, A generalization of Sturmian sequences; combinatorial structure and transcendence, Acta Arith., to appear.
Article |  Zbl 0953.11007
[30] N. WOZNY, L.Q. ZAMBONI, Frequencies of factors in Arnoux-Rauzy sequences, in course of acceptation by Acta Arith.  Zbl 0973.11030
[31] L.Q. ZAMBONI, Une généralisation du théorème de Lagrange sur le développement en fraction continue, C.R. Acad. Sci. Paris, Série I, 327 (1998), 527-530.  MR 2000c:11016 |  Zbl 01223347
Copyright Cellule MathDoc 2005 | Credit | Site Map