The orbit SL(2, ℤ)x is dense in ℝ2 when the initial point x∈ℝ2 has irrational slope. We refine this result from a diophantine perspective. For any target point y∈ℝ2, we introduce two exponents μ(x, y) and μ̂(x, y) that measure the approximation to y by elements γ x of the orbit in terms of the size of γ. We estimate both exponents under various conditions. Our results are optimal when the slope of the target point y is a rational number. In that case we express μ(x, y) and μ̂(x, y) in terms of the irrationality measure of the slope of x.

Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles Annales scientifiques de l'É.N.S. 4 e série, tome 26, n o 6 (1993), p. 645-664 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1993, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Such numerous distinct expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way than what we call "iteration sequences". We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of non-occurrence of certain finite blocks, and the sequences involved in continued fraction expansions. The results are also applied to describe a class of binary quadratic forms with complex coefficients whose values over the set of pairs of Gaussian integers form a dense set of complex numbers.

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Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.