no. 4
Structure of three interval exchange transformations I: an arithmetic study
[Structure de trois transformations d'échanges d'intervalles I : une étude arithmétique]
Ferenczi, Sébastien1 ; Holton, Charles2 ; Zamboni, Luca Q.3
1 Laboratoire de Mathématiques et Physique Théorique, CNRS, UPRES-A 6083, Parc de Grandmont, 37200 Tours (France)
2 University of California, Department of Mathematics, Berkeley CA 94720-3840 (USA)
3 University of North Texas, Department of Mathematics, Denton TX 76203-5116 (USA)
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 : 10.5802/aif.1839
Classification : 11J70, 11J13, 37A05
Keywords: Generalized continued fraction, interval exchange transformations
Mot clés : fractions continues généralisées, échanges d'intervalles

Ferenczi, Sébastien 1 ; Holton, Charles 2 ; Zamboni, Luca Q. 3

1 Laboratoire de Mathématiques et Physique Théorique, CNRS, UPRES-A 6083, Parc de Grandmont, 37200 Tours (France)
2 University of California, Department of Mathematics, Berkeley CA 94720-3840 (USA)
3 University of North Texas, Department of Mathematics, Denton TX 76203-5116 (USA) 
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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) no. 4, pp. 861-901. doi : 10.5802/aif.1839. https://aif.centre-mersenne.org/articles/10.5802/aif.1839/

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