Nous généralisons en dimension deux la méthode de singularisation développée par C. Kraikamp au cours des années 90 dans ses travaux sur les systèmes dynamiques associées aux fractions continues, en relation avec certaines propriétés d’approximations diophantiennes. Nous appliquons la méthode à l’algorithme de Brun en dimension 2 et montrons comment utiliser cette technique et d’autres analogues pour transférer des propriétés métriques et diophantiennes d’un algorithme à l’autre. Une conséquence de cette étude est la construction d’un algorithme qui améliore les propriétés d’approximations par comparaisons avec celles de l’algorithme de Brun.
The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In particular, we are interested in the transferability of the density of the invariant measure. Finally, we use this method to construct an algorithm which improves approximation properties, as opposed to Brun’s algorithm.
@article{JTNB_2004__16_3_705_0,
author = {Schratzberger, Bernhard},
title = {S-expansions in dimension two},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {705--732},
publisher = {Universit\'e Bordeaux 1},
volume = {16},
number = {3},
year = {2004},
doi = {10.5802/jtnb.467},
zbl = {1073.11049},
mrnumber = {2144964},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.467/}
}
TY - JOUR AU - Schratzberger, Bernhard TI - S-expansions in dimension two JO - Journal de théorie des nombres de Bordeaux PY - 2004 SP - 705 EP - 732 VL - 16 IS - 3 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.467/ DO - 10.5802/jtnb.467 LA - en ID - JTNB_2004__16_3_705_0 ER -
Schratzberger, Bernhard. S-expansions in dimension two. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 705-732. doi : 10.5802/jtnb.467. http://www.numdam.org/articles/10.5802/jtnb.467/
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