Weak mixing and eigenvalues for Arnoux-Rauzy sequences
[Mélange faible et valeurs propres pour les suites d’Arnoux-Rauzy]
Cassaigne, Julien ; Ferenczi, Sébastien ; Messaoudi, Ali
Annales de l'Institut Fourier, Tome 58 (2008), p. 1983-2005 / Harvested from Numdam

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.

Publié le : 2008-01-01
DOI : https://doi.org/10.5802/aif.2403
Classification:  37B10,  68R15
Mots clés: dynamique symbolique, complexité, valeurs propres
@article{AIF_2008__58_6_1983_0,
     author = {Cassaigne, Julien and Ferenczi, S\'ebastien and Messaoudi, Ali},
     title = {Weak mixing and eigenvalues for Arnoux-Rauzy sequences},
     journal = {Annales de l'Institut Fourier},
     volume = {58},
     year = {2008},
     pages = {1983-2005},
     doi = {10.5802/aif.2403},
     zbl = {1151.37013},
     mrnumber = {2473626},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_6_1983_0}
}
Cassaigne, Julien; Ferenczi, Sébastien; Messaoudi, Ali. Weak mixing and eigenvalues for Arnoux-Rauzy sequences. Annales de l'Institut Fourier, Tome 58 (2008) pp. 1983-2005. doi : 10.5802/aif.2403. http://gdmltest.u-ga.fr/item/AIF_2008__58_6_1983_0/

[1] Arnoux, Pierre Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore, Bull. Soc. Math. France, Tome 116 (1988) no. 4, pp. 489-500 | Numdam | Zbl 0703.58045

[2] Arnoux, Pierre; Ito, Shunji Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Tome 8 (2001) no. 2, pp. 181-207 (Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000)) | MR 1838930 | Zbl 1007.37001

[3] Arnoux, Pierre; Rauzy, Gérard Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France, Tome 119 (1991) no. 2, pp. 199-215 | Numdam | MR 1116845 | Zbl 0789.28011

[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, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 385 (2005), pp. 333-364 | MR 2180244 | Zbl pre02236674

[5] Boshernitzan, Michael A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math., Tome 44 (1984/85), pp. 77-96 | Article | MR 801288 | Zbl 0602.28008

[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), Tome 72 (2005) no. 3, pp. 799-816 | Article | MR 2190338 | Zbl 1095.54016

[7] Cassaigne, J.; Ferenczi, S.; Zamboni, L. Q. Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 4, pp. 1265-1276 | Article | Numdam | MR 1799745 | Zbl 1004.37008

[8] Chekhova, Nataliya; Hubert, Pascal; Messaoudi, Ali Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux, Tome 13 (2001) no. 2, pp. 371-394 | Article | Numdam | MR 1879664 | Zbl 1038.37010

[9] Choksi, J. R.; Nadkarni, M. G. The group of eigenvalues of a rank one transformation, Canad. Math. Bull., Tome 38 (1995) no. 1, pp. 42-54 | Article | MR 1319899 | Zbl 0833.28008

[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), Tome 67 (2003) no. 3, pp. 790-804 | Article | MR 1967706 | Zbl 1045.54011

[11] Coven, Ethan M.; Hedlund, G. A. Sequences with minimal block growth, Math. Systems Theory, Tome 7 (1973), pp. 138-153 | Article | MR 322838 | Zbl 0256.54028

[12] Ferenczi, Sébastien Systems of finite rank, Colloq. Math., Tome 73 (1997) no. 1, pp. 35-65 | MR 1436950 | Zbl 0883.28014

[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., Tome 62 (1940), pp. 1-42 | Article | MR 745 | Zbl 0022.34003

[15] Oren, Ishai Admissible functions with multiple discontinuities, Israel J. Math., Tome 42 (1982) no. 4, pp. 353-360 | Article | MR 682320 | Zbl 0533.28009

[16] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Tome 110 (1982) no. 2, pp. 147-178 | Numdam | MR 667748 | Zbl 0522.10032