Weak mixing and eigenvalues for Arnoux-Rauzy sequences

Cassaigne, Julien; Ferenczi, Sébastien; Messaoudi, Ali

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1983-2005.

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.

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.

[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