no. 3
On the closedness of approximation spectra
Parkkonen, Jouni1 ; Paulin, Frédéric2
1 Department of Mathematics and Statistics P.O. Box 35 40014 University of Jyväskylä, FINLAND
2 Département de Mathématique et Applications, UMR 8553 CNRS École Normale Supérieure 45 rue d’Ulm 75230 PARIS Cedex 05, FRANCE
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 703-712.

Le spectre classique de Lagrange pour l’approximation des nombres réels par des rationnels, est fermé, par un théorème de Cusick. Plus généralement, nous montrons que de nombreux spectres d’approximation sont fermés, en utilisant des propriétés de pénétration du flot géodésique dans des voisinages de pointes de variétés à courbure strictement négative, et un résultat de Maucourant [Mau].

Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

DOI : 10.5802/jtnb.696
Parkkonen, Jouni 1 ; Paulin, Frédéric 2

1 Department of Mathematics and Statistics P.O. Box 35 40014 University of Jyväskylä, FINLAND
2 Département de Mathématique et Applications, UMR 8553 CNRS École Normale Supérieure 45 rue d’Ulm 75230 PARIS Cedex 05, FRANCE 
@article{JTNB_2009__21_3_703_0,
     author = {Parkkonen, Jouni and Paulin, Fr\'ed\'eric},
     title = {On the closedness of approximation spectra},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {703--712},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.696},
     zbl = {1205.11083},
     mrnumber = {2605541},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.696/}
}
TY  - JOUR
AU  - Parkkonen, Jouni
AU  - Paulin, Frédéric
TI  - On the closedness of approximation spectra
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 703
EP  - 712
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.696/
DO  - 10.5802/jtnb.696
LA  - en
ID  - JTNB_2009__21_3_703_0
ER  - 
%0 Journal Article
%A Parkkonen, Jouni
%A Paulin, Frédéric
%T On the closedness of approximation spectra
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 703-712
%V 21
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.696/
%R 10.5802/jtnb.696
%G en
%F JTNB_2009__21_3_703_0
Parkkonen, Jouni; Paulin, Frédéric. On the closedness of approximation spectra. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 703-712. doi : 10.5802/jtnb.696. http://archive.numdam.org/articles/10.5802/jtnb.696/

[Bor] A. Borel, Reduction theory for arithmetic groups. In “Algebraic Groups and Discontinuous Subgroups”, A. Borel and G. D. Mostow eds, Amer. Math. Soc. 1966. | MR | Zbl

[BHC] A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups. Ann. of Math. 75 (1962), 485–535. | MR | Zbl

[Bow] B. Bowditch, Geometrical finiteness for hyperbolic groups. J. Funct. Ana. 113 (1993), 245–317. | MR | Zbl

[BH] M. R. Bridson, A. Haefliger, Metric spaces with non-positive curvature. Grund. math. Wiss. 319, Springer Verlag (1998). | MR | Zbl

[BK] P. Buser, H. Karcher, Gromov’s almost flat manifolds. Astérisque 81, Soc. Math. France, 1981. | Numdam | MR | Zbl

[CF] T. Cusick, M. Flahive, The Markoff and Lagrange spectra. Math. Surv. Mono. 30. Amer. Math. Soc. 1989. | MR | Zbl

[EGM] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory. Springer Mono. Math., Springer-Verlag, 1998. | MR | Zbl

[Gol] W. M. Goldman, Complex hyperbolic geometry. Oxford Univ. Press, 1999. | MR | Zbl

[HP1] S. Hersonsky, F. Paulin, Diophantine approximation for negatively curved manifolds. Math. Zeit. 241 (2002) 181–226. | MR | Zbl

[HP2] S. Hersonsky, F. Paulin, Diophantine Approximation on Negatively Curved Manifolds and in the Heisenberg Group. In “Rigidity in dynamics and geometry” (Cambridge, 2000), M. Burger, A. Iozzi eds, Springer Verlag (2002), 203–226. | Zbl

[Kel] R. Kellerhals, Quaternions and some global properties of hyperbolic 5-manifolds, Canad. J. Math. 55 (2003) 1080–1099. | MR | Zbl

[Mau] F. Maucourant, Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques. Erg. Theo. Dyn. Sys. 23 (2003), 193–205. | MR | Zbl

[PP1] J. Parkkonen, F. Paulin, Appendix: Diophantine Approximation on Hyperbolic Surfaces. In “Rigidity in dynamics and geometry” (Cambridge, 2000), M. Burger, A. Iozzi eds, Springer Verlag (2002), 227–236. | MR | Zbl

[PP2] J. Parkkonen, F. Paulin, Sur les rayons de Hall en approximation diophantienne. Comptes Rendus Math. 344 (2007), 611–614. | MR | Zbl

[PP3] J. Parkkonen, F. Paulin, Prescribing the behaviour of geodesics in negative curvature. Geometry & Topology 14 (2010), 277–392.

[Vig] M. F. Vigneras, Arithmétique des algèbres de quaternions. Lect. Notes 800, Springer Verlag, 1980. | MR | Zbl

Cité par Sources :