A Rauzy fractal unbounded in all directions of the plane

Mélodie Andrieu

doi:10.5802/crmath.162

Combinatoire, Systèmes dynamiques

A Rauzy fractal unbounded in all directions of the plane

Mélodie Andrieu ¹

¹ Institut de Mathématiques de Marseille, I2M, Marseille, France

Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 399-407.

Résumé
Abstract

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.

Reçu le : 2020-09-15
Révisé le : 2020-11-01
Accepté le : 2020-11-29
Publié le : 2021-05-27

MR Zbl

DOI : 10.5802/crmath.162

Affiliations des auteurs :

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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     title = {A {Rauzy} fractal unbounded in all directions of the plane},
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     pages = {399--407},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
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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/

Bibliographie
Cité par

[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 $2 n + 1$ , 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

Mélodie Andrieu; Léo Vivion Minimal Complexities for Infinite Words Written with d Letters, Combinatorics on Words, Volume 13899 (2023), p. 3 | DOI:10.1007/978-3-031-33180-0_1

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