Soient un espace symétrique de type non compact et un groupe discret d’isométries de du type de Schottky. Dans cet article, nous donnons des équivalents des fonctions orbitales de comptage pour l’action de sur .
Let be a symmetric space of noncompact type and a discrete group of isometries of of Schottky type. In this paper, we give asymptotics of the orbitals counting functions associated to the action of on .
Classification : 22E40, 53C35, 37D35
Mots clés : groupes de Lie, sous-groupes discrets, géométrie en rang supérieur, formalisme thermodynamique.
@article{AIF_2005__55_2_373_0, author = {Quint, Jean-Fran\c cois}, title = {Groupes de Schottky et comptage}, journal = {Annales de l'Institut Fourier}, pages = {373--429}, publisher = {Association des Annales de l'institut Fourier}, volume = {55}, number = {2}, year = {2005}, doi = {10.5802/aif.2102}, zbl = {1087.22010}, mrnumber = {2147895}, language = {fr}, url = {http://www.numdam.org/item/AIF_2005__55_2_373_0/} }
Quint, Jean-François. Groupes de Schottky et comptage. Annales de l'Institut Fourier, Tome 55 (2005) no. 2, pp. 373-429. doi : 10.5802/aif.2102. http://www.numdam.org/item/AIF_2005__55_2_373_0/
[16 Mesures de Patterson-Sullivan en rang supérieur, Geometric and functional analysis, Volume 12 (2002), pp. 776-809 | Article | MR 1935549 | Zbl 02184586
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[2] Lalley's theorem on periodic orbits of hyperbolic flows, Ergodic theory and dynamical systems, Volume 18 (1998), pp. 17-39 | MR 1609507 | Zbl 0915.58074
[3] Propriétés asymptotiques des groupes linéaires, Geometric and functional analysis, Volume 7 (1997), pp. 1-47 | Article | MR 1437472 | Zbl 0947.22003
[4] Propriétés asymptotiques des groupes linéaires (II), Advanced studies in pure mathematics, Volume 26 (2000), pp. 33-48 | MR 1770716 | Zbl 0960.22012
[5] Limit sets of groups of linear transformations (Sankhya Series A), Volume 62 (2000), pp. 367-385 | Zbl 01644967
[6] Prevalence of rapid mixing in hyperbolic flows, Ergodic theory and dynamical systems, Volume 18 (1998), pp. 1097-1114 | Article | MR 1653299 | Zbl 0918.58058
[7] Mixing, counting and equidistribution in Lie groups, Duke mathematical journal, Volume 71 (1993), pp. 181-209 | Article | MR 1230290 | Zbl 0798.11025
[8] Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press, San Diego, 1978 | MR 514561 | Zbl 0451.53038
[9] Closed orbits in homology classes, Publications mathématiques de l'IHES, Volume 71 (1990), pp. 5-32 | Numdam | MR 1079641 | Zbl 0728.58026
[10] Renewal theorems in symbolic dynamics, with application to geodesic flows, noneuclidean tessellations and their fractal limits, Acta mathematica, Volume 163 (1989), pp. 1-55 | Article | MR 1007619 | Zbl 0701.58021
[11] -regular elements in Zariski dense subgroups, Oxford quaterly journal of mathematics, Volume 45 (1994), pp. 541-545 | MR 1315463 | Zbl 0828.22010
[12] Zeta functions and the periodic orbit structure for hyperbolic dynamics (Astérisque) (1990), p. 187-188 | Zbl 0726.58003
[13] Linear actions of free groups, Annales de l'institut Fourier, Volume 51 (2001), pp. 131-150 | Article | Numdam | MR 1821072 | Zbl 0967.37016
[14] Asymptotic expansions for closed orbits in homology classes, GeometriæDedicata, Volume 87 (2001), pp. 123-160 | MR 1866845 | Zbl 1049.37021
[15] Divergence exponentielle des sous-groupes discrets en rang supérieur, Commentarii Mathematicii Helvetici, Volume 77 (2002), pp. 563-608 | Article | MR 1933790 | Zbl 1010.22018
[17] L'indicateur de croissance des groupes de Schottky, Ergodic theory and dynamical systems, Volume 23 (2003), pp. 249-272 | MR 1971205 | Zbl 1037.22024
[18] Ergodicité et équidistribution en courbure négative (Mémoires), Volume 95 (2003) | Numdam | Zbl 1056.37034
[18] The infinite word problem and limit sets in Fuchsian groups, Ergodic theory and dynamical systems, Volume 1 (1981), pp. 337-360 | MR 662473 | Zbl 0483.30029
[19] Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, Journal f\" ur die reine und angewandte Mathematik, Volume 247 (1971), pp. 196-220 | MR 277536 | Zbl 0227.20015