no. 4
Combinatoire, Systèmes dynamiques
A Rauzy fractal unbounded in all directions of the plane
Andrieu, Mélodie1
1 Institut de Mathématiques de Marseille, I2M, Marseille, France
Comptes Rendus. Mathématique, Tome 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 ℤ 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 ℤ 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
Andrieu, Mélodie 1

1 Institut de Mathématiques de Marseille, I2M, Marseille, France 
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Andrieu, Mélodie. A Rauzy fractal unbounded in all directions of the plane. Comptes Rendus. Mathématique, Tome 359 (2021) no. 4, pp. 399-407. doi : 10.5802/crmath.162. http://archive.numdam.org/articles/10.5802/crmath.162/

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

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

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

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

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

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

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

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

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

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