Comptes Rendus
Combinatoire, Systèmes dynamiques
A Rauzy fractal unbounded in all directions of the plane
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 399-407.

Nous construisons explicitement un mot d’Arnoux–Rauzy pour lequel l’ensemble des différences possibles des facteurs abélianisés est égal à 3 . En particulier, le déséquilibre de ce mot est infini, et son fractal de Rauzy n’est borné dans aucune direction du plan.

We construct an Arnoux–Rauzy word for which the set of all differences of two abelianized factors is equal to 3 . In particular, the imbalance of this word is infinite – and its Rauzy fractal is unbounded in all directions of the plane.

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DOI : 10.5802/crmath.162
Mélodie Andrieu 1

1 Institut de Mathématiques de Marseille, I2M, Marseille, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Mélodie Andrieu. A Rauzy fractal unbounded in all directions of the plane. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 399-407. doi : 10.5802/crmath.162. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.162/

[1] Mélodie Andrieu Exceptional trajectories in the symbolic dynamics of multidimentional continued fraction algorithms, Ph. D. Thesis, Institut de Mathématiques de Marseille, Marseille, France (2021) | Zbl

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

[3] Pierre Arnoux; Štěpán Starosta The Rauzy Gasket, Further Developments in Fractals and Related Fields. Mathematical foundations and connections (Trends in Mathematics), Springer, 2013, pp. 1-23 | Zbl

[4] Julien Cassaigne; Sébastien Ferenczi; Luca Q. Zamboni Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier, Volume 50 (2000), pp. 1265-1276 | DOI | Numdam | MR | Zbl

[5] Julien Cassaigne; Sébastien Labbé; Julien Leroy A Set of Sequences of Complexity 2n+1, Combinatorics on words. 11th international conference, WORDS 2017 Proceedings (2017), pp. 144-156 | Zbl

[6] Ivan Dynnikov; Pascal Hubert; Alexandra Skripchenko Dynamical systems around the Rauzy gasket and their ergodic properties (2020) (in preparation, https://arxiv.org/abs/2011.15043)

[7] M. Lothaire Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, 17, Cambridge University Press, 1997 (Foreword by Roger Lyndon) | MR | Zbl

[8] Valery Iustinovich Oseledets A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., Volume 19 (1968), pp. 197-231 | Zbl

[9] Fritz Schweiger Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, 2000 | Zbl

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