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

Nous proposons de formaliser une méthode d’approximation diophantienne dans en considérant l’action de PGL 2 () 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.

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.

@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, Tome 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