Dans cet article nous décrivons une généralisation à la dimension de l’ algorithme d’Euclide, qui provient de la dynamique des échanges de 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 .
In this paper we describe a -dimensional generalization of the Euclidean algorithm which stems from the dynamics of -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
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/} }
TY - JOUR AU - Ferenczi, Sébastien AU - Holton, Charles AU - Zamboni, Luca Q. TI - Structure of three interval exchange transformations I: an arithmetic study JO - Annales de l'Institut Fourier PY - 2001 DA - 2001/// SP - 861 EP - 901 VL - 51 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1839/ UR - https://zbmath.org/?q=an%3A1029.11036 UR - https://www.ams.org/mathscinet-getitem?mr=1849209 UR - https://doi.org/10.5802/aif.1839 DO - 10.5802/aif.1839 LA - en ID - AIF_2001__51_4_861_0 ER -
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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