no. 4

Structure of three interval exchange transformations I: an arithmetic study
[Structure de trois transformations d'échanges d'intervalles I : une étude arithmétique]
Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 861-901.

Dans cet article nous décrivons une généralisation à la dimension 2 de l’ algorithme d’Euclide, qui provient de la dynamique des échanges de 3 intervalles. Nous examinons diverses propriétés diophantiennes de cet algorithme, en particulier la qualité de l’approximation simultanée. Nous montrons qu’il vérifie un théorème de type Lagrange : l’algorithme est finalement périodique si et seulement si les paramètres sont dans la même extension quadratique de ℚ.

In this paper we describe a 2-dimensional generalization of the Euclidean algorithm which stems from the dynamics of 3-interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of ℚ.

DOI : https://doi.org/10.5802/aif.1839
Classification : 11J70,  11J13,  37A05
Mots clés : fractions continues généralisées, échanges d'intervalles 
@article{AIF_2001__51_4_861_0,
     author = {Ferenczi, S\'ebastien and Holton, Charles and Zamboni, Luca Q.},
     title = {Structure of three interval exchange transformations I: an arithmetic study},
     journal = {Annales de l'Institut Fourier},
     pages = {861--901},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     doi = {10.5802/aif.1839},
     zbl = {1029.11036},
     mrnumber = {1849209},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1839/}
}
Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three interval exchange transformations I: an arithmetic study. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 861-901. doi : 10.5802/aif.1839. http://www.numdam.org/articles/10.5802/aif.1839/

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