Weak mixing and eigenvalues for Arnoux-Rauzy sequences
Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 1983-2005.

We define by simple conditions two wide subclasses of the so-called Arnoux-Rauzy systems; the elements of the first one share the property of (measure-theoretic) weak mixing, thus we generalize and improve a counter-example to the conjecture that these systems are codings of rotations; those of the second one have eigenvalues, which was known hitherto only for a very small set of examples.

Nous définissons par des condition simples deux larges sous-classes des systèmes dits d’Arnoux-Rauzy ; les membres de la première possèdent la propriété de mélange faible (mesurable), ce qui généralise et améliore un contre-exemple à la conjecture que tous ces systèmes sont des codages de rotations ; ceux de la seconde ont des valeurs propres, ce qui n’était connu jusqu’ici que pour un ensemble très restreint d’exemples.

DOI: 10.5802/aif.2403
Classification: 37B10, 68R15
Keywords: Symbolic dynamics, complexity, eigenvalues
Cassaigne, Julien 1; Ferenczi, Sébastien 1; Messaoudi, Ali 2

1 Institut de Mathématiques de Luminy CNRS-UMR 6206 Case 907, 163 av.  de Luminy 13288 Marseille Cedex 9 (France) and Fédération de Recherche des Unités de Mathématiques de Marseille CNRS-FR 2291
2 UNESP-Universidade Estadual Paulista Rua Cristovo Colombo, 2265 Jardim Nazareth 15054-000 São José do Rio Preto, SP (Brazil)
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Cassaigne, Julien; Ferenczi, Sébastien; Messaoudi, Ali. Weak mixing and eigenvalues for Arnoux-Rauzy sequences. Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 1983-2005. doi : 10.5802/aif.2403. https://aif.centre-mersenne.org/articles/10.5802/aif.2403/

[1] Arnoux, Pierre Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore, Bull. Soc. Math. France, Volume 116 (1988) no. 4, pp. 489-500 | Numdam | Zbl

[2] Arnoux, Pierre; Ito, Shunji Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Volume 8 (2001) no. 2, pp. 181-207 Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000) | MR | Zbl

[3] Arnoux, Pierre; Rauzy, Gérard Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France, Volume 119 (1991) no. 2, pp. 199-215 | Numdam | MR | Zbl

[4] Berthé, Valérie; Ferenczi, Sébastien; Zamboni, Luca Q. Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics (Contemp. Math.), Volume 385, Amer. Math. Soc., Providence, RI, 2005, pp. 333-364 | MR

[5] Boshernitzan, Michael A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math., Volume 44 (1984/85), pp. 77-96 | DOI | MR | Zbl

[6] Bressaud, Xavier; Durand, Fabien; Maass, Alejandro Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2), Volume 72 (2005) no. 3, pp. 799-816 | DOI | MR | Zbl

[7] Cassaigne, J.; Ferenczi, S.; Zamboni, L. Q. Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 4, pp. 1265-1276 | DOI | Numdam | MR | Zbl

[8] Chekhova, Nataliya; Hubert, Pascal; Messaoudi, Ali Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux, Volume 13 (2001) no. 2, pp. 371-394 | DOI | Numdam | MR | Zbl

[9] Choksi, J. R.; Nadkarni, M. G. The group of eigenvalues of a rank one transformation, Canad. Math. Bull., Volume 38 (1995) no. 1, pp. 42-54 | DOI | MR | Zbl

[10] Cortez, Maria Isabel; Durand, Fabien; Host, Bernard; Maass, Alejandro Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2), Volume 67 (2003) no. 3, pp. 790-804 | DOI | MR | Zbl

[11] Coven, Ethan M.; Hedlund, G. A. Sequences with minimal block growth, Math. Systems Theory, Volume 7 (1973), pp. 138-153 | DOI | MR | Zbl

[12] Ferenczi, Sébastien Systems of finite rank, Colloq. Math., Volume 73 (1997) no. 1, pp. 35-65 | MR | Zbl

[13] Ferenczi, Sébastien; Zamboni, Luca Q. Examples of 4 -interval exchange transformations (http://iml.univ-mrs.fr/~ferenczi/fz2.pdf)

[14] Morse, Marston; Hedlund, Gustav A. Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., Volume 62 (1940), pp. 1-42 | DOI | MR | Zbl

[15] Oren, Ishai Admissible functions with multiple discontinuities, Israel J. Math., Volume 42 (1982) no. 4, pp. 353-360 | DOI | MR | Zbl

[16] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Volume 110 (1982) no. 2, pp. 147-178 | Numdam | MR | Zbl

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