Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne; Sébastien Ferenczi; Luca Q. Zamboni

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1265-1276
  • ISSN: 0373-0956

Abstract

top
In a 1982 paper Rauzy showed that the subshift ( X , T ) generated by the morphism 1 12 , 2 13 and 3 1 is a natural coding of a rotation on the two-dimensional torus 𝕋 2 , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in 2 , each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity 2 n + 1 satisfying a combinatorial criterion known as the condition of Arnoux and Rauzy codes the orbit of a point under a rotation on 𝕋 2 . In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence ω * which is unbalanced in the following sense: for each N > 0 there exist two factors of ω * 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 𝕋 2 .

How to cite

top

Cassaigne, Julien, Ferenczi, Sébastien, and Zamboni, Luca Q.. "Imbalances in Arnoux-Rauzy sequences." Annales de l'institut Fourier 50.4 (2000): 1265-1276. <http://eudml.org/doc/75456>.

@article{Cassaigne2000,
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 $N&gt;0$ 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 $\{\Bbb T\}^2$.},
author = {Cassaigne, Julien, Ferenczi, Sébastien, Zamboni, Luca Q.},
journal = {Annales de l'institut Fourier},
keywords = {infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences},
language = {eng},
number = {4},
pages = {1265-1276},
publisher = {Association des Annales de l'Institut Fourier},
title = {Imbalances in Arnoux-Rauzy sequences},
url = {http://eudml.org/doc/75456},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Ferenczi, Sébastien
AU - Zamboni, Luca Q.
TI - Imbalances in Arnoux-Rauzy sequences
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1265
EP - 1276
AB - 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 $N&gt;0$ 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 ${\Bbb T}^2$.
LA - eng
KW - infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences
UR - http://eudml.org/doc/75456
ER -

References

top
  1. [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. Zbl0703.58045MR91a:58138
  2. [2] P. ARNOUX, V. BERHÉ, S. ITO, Discrete planes, ℤ2-actions, Jacobi-Perron algorithm and substitutions, preprint (1999). 
  3. [3] P. ARNOUX, S. ITO, Pisot substitutions and Rauzy fractals, preprint 98-18, Institut de Mathématiques de Luminy, Marseille (1998). Zbl1007.37001
  4. [4] P. ARNOUX, G. RAUZY, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France, 119 (1991), 199-215. Zbl0789.28011MR92k:58072
  5. [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). Zbl0970.68124
  6. [6] V. CANTERINI, A. SIEGEL, Geometric representations of Pisot substitutions, preprint (1999). Zbl1142.37302
  7. [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. Zbl0916.68114MR2000c:68110
  8. [8] R. V. CHACON, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562. Zbl0186.37203MR40 #297
  9. [9] N. CHEKHOVA, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une question d'Hedlund et Morse, preprint (1999). 
  10. [10] N. CHEKHOVA, Algorithme d'approximation et propriétés ergodiques des suites d'Arnoux-Rauzy, preprint (1999). 
  11. [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. [12] X. DROUBAY, J. JUSTIN, G. PIRILLO, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comp. Sci., to appear. Zbl0981.68126
  13. [13] S. FERENCZI, Bounded remainder sets, Acta Arith., 61 (1992), 319-326. Zbl0774.11037MR93f:11059
  14. [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. Zbl0855.28008MR96m:28018
  15. [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). Zbl1029.11036
  16. [16] S. FERENCZI, C. HOLTON, L.Q. ZAMBONI, Structure of three interval exchange transformations II: A combinatorial study; ergodic and spectral properties, preprint (2000). Zbl1029.11036
  17. [17] S. FERENCZI, C. MAUDUIT, Transcendence of numbers with a low complexity expansion, J. Number Theory, 67 (1997), 146-161. Zbl0895.11029MR98m:11079
  18. [18] S. ITO, M. KIMURA, On Rauzy fractal, Japan J. Indust. Appl. Math., 8 (1991), 461-486. Zbl0734.28010MR93d:11084
  19. [19] H. KESTEN, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith., 12 (1966), 193-212. Zbl0144.28902MR35 #155
  20. [20] M. LOTHAIRE, Algebraic Combinatorics on Words, Chapter 2: Sturmian words, by J. Berstel, P. Séébold, to appear. Zbl1001.68093
  21. [21] A. MESSAOUDI, Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Th. Nombres de Bordeaux, 10 (1998), 135-162. Zbl0918.11048MR2002c:11091
  22. [22] A. MESSAOUDI, Frontière du fractal de Rauzy et système de numération complexe, Acta Arith., to appear. Zbl0968.28005
  23. [23] M. MORSE, G.A. HEDLUND, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866. Zbl0019.33502JFM64.0798.04
  24. [24] M. MORSE, G.A. HEDLUND, Symbolic dynamics II: Sturmian sequences, Amer. J. Math., 62 (1940), 1-42. Zbl0022.34003MR1,123dJFM66.0188.03
  25. [25] G. RAUZY, Une généralisation du développement en fraction continue, Séminaire Delange - Pisot - Poitou, 1976-1977, Paris, exposé 15, 15-01-15-16. Zbl0369.28015MR83e:10077
  26. [26] G. RAUZY, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. Zbl0414.28018MR82m:10076
  27. [27] G. RAUZY, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178. Zbl0522.10032MR84h:10074
  28. [28] G. RAUZY, Ensembles à restes bornés, Séminaire de Théorie des Nombres de Bordeaux, 1983-1984, exposé 24, 24-01-24-12. Zbl0547.10044MR86g:28024
  29. [29] R. RISLEY, L.Q. ZAMBONI, A generalization of Sturmian sequences; combinatorial structure and transcendence, Acta Arith., to appear. Zbl0953.11007
  30. [30] N. WOZNY, L.Q. ZAMBONI, Frequencies of factors in Arnoux-Rauzy sequences, in course of acceptation by Acta Arith. Zbl0973.11030
  31. [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. Zbl1039.11500MR2000c:11016

Citations in EuDML Documents

top
  1. Christiane Frougny, Zuzana Masáková, Edita Pelantová, Complexity of infinite words associated with beta-expansions
  2. Christiane Frougny, Zuzana Masáková, Edita Pelantová, Complexity of infinite words associated with beta-expansions
  3. Ondřej Turek, Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
  4. Julien Bernat, Study of irreducible balanced pairs for substitutive languages
  5. Julien Cassaigne, Sébastien Ferenczi, Ali Messaoudi, Weak mixing and eigenvalues for Arnoux-Rauzy sequences
  6. Boris Adamczewski, Codages de rotations et phénomènes d'autosimilarité
  7. Valérie Berthé, Timo Jolivet, Anne Siegel, Connectedness of fractals associated with Arnoux–Rauzy substitutions
  8. Sébastien Ferenczi, Charles Holton, Luca Q. Zamboni, Structure of three interval exchange transformations I: an arithmetic study
  9. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets
  10. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets

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.