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Jouni Parkkonen; Frédéric Paulin
On the closedness of approximation spectra
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 703-712
Article: subscription required
[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 204533 | Zbl 0213.47201
[BHC] A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups. Ann. of Math. 75 (1962), 485–535. MR 147566 | Zbl 0107.14804
[Bow] B. Bowditch, Geometrical finiteness for hyperbolic groups. J. Funct. Ana. 113 (1993), 245–317. MR 1218098 | Zbl 0789.57007
[BH] M. R. Bridson, A. Haefliger, Metric spaces with non-positive curvature. Grund. math. Wiss. 319, Springer Verlag (1998). MR 1744486 | Zbl 0988.53001
[BK] P. Buser, H. Karcher, Gromov’s almost flat manifolds. Astérisque 81, Soc. Math. France, 1981. MR 619537 | Zbl 0459.53031
[CF] T. Cusick, M. Flahive, The Markoff and Lagrange spectra. Math. Surv. Mono. 30. Amer. Math. Soc. 1989. MR 1010419 | Zbl 0685.10023
[EGM] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory. Springer Mono. Math., Springer-Verlag, 1998. MR 1483315 | Zbl 0888.11001
[Gol] W. M. Goldman, Complex hyperbolic geometry. Oxford Univ. Press, 1999. MR 1695450 | Zbl 0939.32024
[HP1] S. Hersonsky, F. Paulin, Diophantine approximation for negatively curved manifolds. Math. Zeit. 241 (2002) 181–226. MR 1930990 | Zbl 1037.53020
[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 1064.11057
[Kel] R. Kellerhals, Quaternions and some global properties of hyperbolic $5$-manifolds, Canad. J. Math. 55 (2003) 1080–1099. MR 2005283 | Zbl 1054.57019
[Mau] F. Maucourant, Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques. Erg. Theo. Dyn. Sys. 23 (2003), 193–205. MR 1971202 | Zbl 1049.11068
[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 1919403 | Zbl 1064.11058
[PP2] J. Parkkonen, F. Paulin, Sur les rayons de Hall en approximation diophantienne. Comptes Rendus Math. 344 (2007), 611–614. MR 2334070 | Zbl 1151.11030
[PP3] J. Parkkonen, F. Paulin, Prescribing the behaviour of geodesics in negative curvature. Geometry & Topology 14 (2010), 277–392. Zbl pre05624192
[Vig] M. F. Vigneras, Arithmétique des algèbres de quaternions. Lect. Notes 800, Springer Verlag, 1980. MR 580949 | Zbl 0422.12008
Jouni Parkkonen; Frédéric Paulin
On the closedness of approximation spectra
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 703-712
Article: subscription required
Résumé - Abstract
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].
Bibliography
[BHC] A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups. Ann. of Math. 75 (1962), 485–535. MR 147566 | Zbl 0107.14804
[Bow] B. Bowditch, Geometrical finiteness for hyperbolic groups. J. Funct. Ana. 113 (1993), 245–317. MR 1218098 | Zbl 0789.57007
[BH] M. R. Bridson, A. Haefliger, Metric spaces with non-positive curvature. Grund. math. Wiss. 319, Springer Verlag (1998). MR 1744486 | Zbl 0988.53001
[BK] P. Buser, H. Karcher, Gromov’s almost flat manifolds. Astérisque 81, Soc. Math. France, 1981. MR 619537 | Zbl 0459.53031
[CF] T. Cusick, M. Flahive, The Markoff and Lagrange spectra. Math. Surv. Mono. 30. Amer. Math. Soc. 1989. MR 1010419 | Zbl 0685.10023
[EGM] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory. Springer Mono. Math., Springer-Verlag, 1998. MR 1483315 | Zbl 0888.11001
[Gol] W. M. Goldman, Complex hyperbolic geometry. Oxford Univ. Press, 1999. MR 1695450 | Zbl 0939.32024
[HP1] S. Hersonsky, F. Paulin, Diophantine approximation for negatively curved manifolds. Math. Zeit. 241 (2002) 181–226. MR 1930990 | Zbl 1037.53020
[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 1064.11057
[Kel] R. Kellerhals, Quaternions and some global properties of hyperbolic $5$-manifolds, Canad. J. Math. 55 (2003) 1080–1099. MR 2005283 | Zbl 1054.57019
[Mau] F. Maucourant, Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques. Erg. Theo. Dyn. Sys. 23 (2003), 193–205. MR 1971202 | Zbl 1049.11068
[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 1919403 | Zbl 1064.11058
[PP2] J. Parkkonen, F. Paulin, Sur les rayons de Hall en approximation diophantienne. Comptes Rendus Math. 344 (2007), 611–614. MR 2334070 | Zbl 1151.11030
[PP3] J. Parkkonen, F. Paulin, Prescribing the behaviour of geodesics in negative curvature. Geometry & Topology 14 (2010), 277–392. Zbl pre05624192
[Vig] M. F. Vigneras, Arithmétique des algèbres de quaternions. Lect. Notes 800, Springer Verlag, 1980. MR 580949 | Zbl 0422.12008
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