no. 1
Fractions continues hermitiennes et billard hyperbolique
Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 1-15.

The purpose of this paper is to describe a dynamical system (X,T) associated to the Hermite algorithm for the continued fraction expansion of real numbers. It is related to trajectories in hyperbolic billiards. We prove the ergodicity of T and we deduce some results.

Nous proposons de formaliser une méthode d’approximation diophantienne dans ℝ en considérant l’action de PGL 2 (ℤ) sur le demi-plan complexe. On retrouvera le thème classique de la connexion entre développement en fractions continues et flots géodésiques modélisé ici par un billard hyperbolique.

@article{JTNB_1998__10_1_1_0,
     author = {Meignen, Pierrick},
     title = {Fractions continues hermitiennes et billard hyperbolique},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1--15},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {1},
     year = {1998},
     mrnumber = {1827282},
     zbl = {0927.11045},
     language = {fr},
     url = {http://archive.numdam.org/item/JTNB_1998__10_1_1_0/}
}
TY  - JOUR
AU  - Meignen, Pierrick
TI  - Fractions continues hermitiennes et billard hyperbolique
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1998
SP  - 1
EP  - 15
VL  - 10
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_1998__10_1_1_0/
LA  - fr
ID  - JTNB_1998__10_1_1_0
ER  - 
%0 Journal Article
%A Meignen, Pierrick
%T Fractions continues hermitiennes et billard hyperbolique
%J Journal de théorie des nombres de Bordeaux
%D 1998
%P 1-15
%V 10
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_1998__10_1_1_0/
%G fr
%F JTNB_1998__10_1_1_0
Meignen, Pierrick. Fractions continues hermitiennes et billard hyperbolique. Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 1-15. http://archive.numdam.org/item/JTNB_1998__10_1_1_0/

[1] R. Adler, L. Flatto, Cross section maps for geodesic flows, Ergodic Theory and Dynamical Systems, Progress in Math. 2, (ed. A. Katok, Birkhäuser, Boston, 1980), 103-161. | MR | Zbl

[2] N. Bourbaki, Groupes et algèbres de Lie, Ch. 4-6, Hermann, Paris, 1968. | MR | Zbl

[3] L.R. Ford, Rational approximations to irrational complex numbers, Trans. Amer. Math. Soc. 19 (1918), 1-42. | JFM | MR

[4] E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863-877. | MR | Zbl

[5] G. Humbert, Sur la méthode d'approximation d'Hermite, Journal de Maths, 2, (1916), 79-103. | JFM | Numdam

[6] P. Meignen, Groupes de Coxeter et approximation diophantienne, Thèse de l'Université de Caen, France, 1995.

[7] P. Meignen, Generating series for the Coxeter groups and applications, à paraître dans Contributions to Algebra and Geometry. | MR | Zbl

[8] R. Moeckel, Geodesics on modular surfaces and continued fractions, Ergo. Th. and Dynam. Sys 2 (1982), 69-84. | MR | Zbl

[9] C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergo. Th. and Dynam. Sys 6 (1986), 601-625. | MR | Zbl

[10] C. Series, The modular surface and continued fractions, J. London Math. Soc. 31 (1985), 69-80. | MR | Zbl

[11] Ya. G. Sinai, Dynamical systems II, Encyclopedia of Maths. Sciences, 2, Springer-Verlag, Berlin Heidelberg, 1989. | Zbl

[12] Ya. G. Sinai, Geodesic flows on manifolds of constant negative curvature, Dokl. Akad. Nauk. SSSR 131 (1960), 752-755; Soviet Math. Dokl. 1 (1960), 335-339. | MR | Zbl

[13] E.B. Vinberg, Geometry II, Encyclopedia of Maths. Sciences, 29, Springer-Verlag, Berlin Heidelberg, 1993. | MR | Zbl