Structure of three interval exchange transformations I: an arithmetic study

Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q.

Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q.

Annales de l'Institut Fourier, Volume 51 (2001) no. 4, p. 861-901

Abstract
Résumé

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 $ℚ .$

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 $ℚ$ .

MR 1849209 | Zbl 1029.11036 | 5 citations in Numdam

DOI : https://doi.org/10.5802/aif.1839
Classification: 11J70, 11J13, 37A05
Keywords: Generalized continued fraction, interval exchange transformations

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     pages = {861-901},
     doi = {10.5802/aif.1839},
     zbl = {1029.11036},
     mrnumber = {1849209},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_4_861_0}
}

Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three interval exchange transformations I: an arithmetic study. Annales de l'Institut Fourier, Volume 51 (2001) no. 4, pp. 861-901. doi : 10.5802/aif.1839. http://www.numdam.org/item/AIF_2001__51_4_861_0/

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