We construct an Arnoux–Rauzy word for which the set of all differences of two abelianized factors is equal to . 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 . En particulier, le déséquilibre de ce mot est infini, et son fractal de Rauzy n’est borné dans aucune direction du plan.
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.162
@article{CRMATH_2021__359_4_399_0, author = {Andrieu, M\'elodie}, title = {A {Rauzy} fractal unbounded in all directions of the plane}, journal = {Comptes Rendus. Math\'ematique}, pages = {399--407}, publisher = {Acad\'emie des sciences, Paris}, volume = {359}, number = {4}, year = {2021}, doi = {10.5802/crmath.162}, mrnumber = {4264322}, zbl = {07362160}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/crmath.162/} }
TY - JOUR AU - Andrieu, Mélodie TI - A Rauzy fractal unbounded in all directions of the plane JO - Comptes Rendus. Mathématique PY - 2021 SP - 399 EP - 407 VL - 359 IS - 4 PB - Académie des sciences, Paris UR - http://archive.numdam.org/articles/10.5802/crmath.162/ DO - 10.5802/crmath.162 LA - en ID - CRMATH_2021__359_4_399_0 ER -
%0 Journal Article %A Andrieu, Mélodie %T A Rauzy fractal unbounded in all directions of the plane %J Comptes Rendus. Mathématique %D 2021 %P 399-407 %V 359 %N 4 %I Académie des sciences, Paris %U http://archive.numdam.org/articles/10.5802/crmath.162/ %R 10.5802/crmath.162 %G en %F CRMATH_2021__359_4_399_0
Andrieu, Mélodie. 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. http://archive.numdam.org/articles/10.5802/crmath.162/
[1] Exceptional trajectories in the symbolic dynamics of multidimentional continued fraction algorithms, Ph. D. Thesis, Institut de Mathématiques de Marseille, Marseille, France (2021)
[2] Représentation géométrique de suites de complexité , Bull. Soc. Math. Fr., Volume 119 (1991) no. 2, pp. 199-215 | DOI | Numdam | Zbl
[3] The Rauzy Gasket, Further Developments in Fractals and Related Fields. Mathematical foundations and connections (Trends in Mathematics), Springer, 2013, pp. 1-23 | Zbl
[4] Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier, Volume 50 (2000), pp. 1265-1276 | DOI | Numdam | MR | Zbl
[5] A Set of Sequences of Complexity 2n+1, Combinatorics on words. 11th international conference, WORDS 2017 Proceedings, Springer (2017), pp. 144-156 | Zbl
[6] Dynamical systems around the Rauzy gasket and their ergodic properties (2020) (in preparation, https://arxiv.org/abs/2011.15043)
[7] Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, 17, Cambridge University Press, 1997 (Foreword by Roger Lyndon) | Zbl
[8] A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc., Volume 19 (1968), pp. 197-231 | Zbl
[9] Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, 2000 | Zbl
Cited by Sources: