Structure of three interval exchange transformations I: an arithmetic study
[Structure de trois transformations d'échanges d'intervalles I : une étude arithmétique]
Ferenczi, Sébastien ; Holton, Charles ; Zamboni, Luca Q.
Annales de l'Institut Fourier, Tome 51 (2001), p. 861-901 / Harvested from Numdam
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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.

Publié le : 2001-01-01
DOI : https://doi.org/10.5802/aif.1839
Classification:  11J70,  11J13,  37A05
Mots clés: fractions continues généralisées, échanges d'intervalles 
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     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},
     volume = {51},
     year = {2001},
     pages = {861-901},
     doi = {10.5802/aif.1839},
     mrnumber = {1849209},
     zbl = {1029.11036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2001__51_4_861_0}
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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) pp. 861-901. doi : 10.5802/aif.1839. http://gdmltest.u-ga.fr/item/AIF_2001__51_4_861_0/

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