Weak mixing and eigenvalues for Arnoux-Rauzy sequences

Julien Cassaigne[1]; Sébastien Ferenczi[1]; Ali Messaoudi[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)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 6, page 1983-2005
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Cassaigne, Julien, Ferenczi, Sébastien, and Messaoudi, Ali. "Weak mixing and eigenvalues for Arnoux-Rauzy sequences." Annales de l’institut Fourier 58.6 (2008): 1983-2005. <http://eudml.org/doc/10367>.

@article{Cassaigne2008,
abstract = {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.},
affiliation = {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; 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; UNESP-Universidade Estadual Paulista Rua Cristovo Colombo, 2265 Jardim Nazareth 15054-000 São José do Rio Preto, SP (Brazil)},
author = {Cassaigne, Julien, Ferenczi, Sébastien, Messaoudi, Ali},
journal = {Annales de l’institut Fourier},
keywords = {Symbolic dynamics; complexity; eigenvalues; symbolic dynamics; Arnoux-Rauzy systems; weak mixing},
language = {eng},
number = {6},
pages = {1983-2005},
publisher = {Association des Annales de l’institut Fourier},
title = {Weak mixing and eigenvalues for Arnoux-Rauzy sequences},
url = {http://eudml.org/doc/10367},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Ferenczi, Sébastien
AU - Messaoudi, Ali
TI - Weak mixing and eigenvalues for Arnoux-Rauzy sequences
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 1983
EP - 2005
AB - 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.
LA - eng
KW - Symbolic dynamics; complexity; eigenvalues; symbolic dynamics; Arnoux-Rauzy systems; weak mixing
UR - http://eudml.org/doc/10367
ER -

References

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  8. Nataliya Chekhova, Pascal Hubert, Ali Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux 13 (2001), 371-394 Zbl1038.37010MR1879664
  9. J. R. Choksi, M. G. Nadkarni, The group of eigenvalues of a rank one transformation, Canad. Math. Bull. 38 (1995), 42-54 Zbl0833.28008MR1319899
  10. Maria Isabel Cortez, Fabien Durand, Bernard Host, Alejandro Maass, Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2) 67 (2003), 790-804 Zbl1045.54011MR1967706
  11. Ethan M. Coven, G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153 Zbl0256.54028MR322838
  12. Sébastien Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), 35-65 Zbl0883.28014MR1436950
  13. Sébastien Ferenczi, Luca Q. Zamboni, Examples of 4 -interval exchange transformations Zbl1147.37008
  14. Marston Morse, Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42 Zbl0022.34003MR745
  15. Ishai Oren, Admissible functions with multiple discontinuities, Israel J. Math. 42 (1982), 353-360 Zbl0533.28009MR682320
  16. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178 Zbl0522.10032MR667748

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