A Rauzy fractal unbounded in all directions of the plane

Mélodie Andrieu

doi:10.5802/crmath.162

Combinatorics, Dynamical systems

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.

Abstract
Résumé

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.

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.

Received: 2020-09-15
Revised: 2020-11-01
Accepted: 2020-11-29
Published online: 2021-05-27

MR Zbl

DOI: 10.5802/crmath.162

Author's affiliations:

Mélodie Andrieu ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {M\'elodie Andrieu},
     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/

References
Cited by

[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

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