Groupes de Schottky et comptage

Quint, Jean-François

Groupes de Schottky et comptage [ Schottky groups and counting ]

Quint, Jean-François

Annales de l'Institut Fourier, Volume 55 (2005) no. 2, p. 373-429

Abstract
Résumé

Let $X$ be a symmetric space of noncompact type and $Γ$ a discrete group of isometries of $X$ of Schottky type. In this paper, we give asymptotics of the orbitals counting functions associated to the action of $Γ$ on $X$ .

Soient $X$ un espace symétrique de type non compact et $Γ$ un groupe discret d’isométries de $X$ du type de Schottky. Dans cet article, nous donnons des équivalents des fonctions orbitales de comptage pour l’action de $Γ$ sur $X$ .

MR 2147895 | Zbl 1087.22010 | 3 citations in Numdam

DOI : https://doi.org/10.5802/aif.2102
Classification: 22E40, 53C35, 37D35
Keywords: Lie groups, discrete subgroups, higher rank geometry, thermodynamical formalism

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {2},
     year = {2005},
     pages = {373-429},
     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, Volume 55 (2005) no. 2, pp. 373-429. doi : 10.5802/aif.2102. http://www.numdam.org/item/AIF_2005__55_2_373_0/

References

[16 J.-F. Quint Mesures de Patterson-Sullivan en rang supérieur, Geometric and functional analysis, Tome 12 (2002), pp. 776-809 | Article | MR 1935549 | Zbl 02184586

[1] N. Anantharaman Precise counting results for closed orbits of Anosov flows, Annales Scientifiques de l'École Normale Supérieure, Tome 33 (2000), pp. 33-56 | Numdam | MR 1743718 | Zbl 0992.37026

[2] M. Babillot; F. Ledrappier Lalley's theorem on periodic orbits of hyperbolic flows, Ergodic theory and dynamical systems, Tome 18 (1998), pp. 17-39 | MR 1609507 | Zbl 0915.58074

[3] Y. Benoist Propriétés asymptotiques des groupes linéaires, Geometric and functional analysis, Tome 7 (1997), pp. 1-47 | Article | MR 1437472 | Zbl 0947.22003

[4] Y. Benoist Propriétés asymptotiques des groupes linéaires (II), Advanced studies in pure mathematics, Tome 26 (2000), pp. 33-48 | MR 1770716 | Zbl 0960.22012

[5] J.-P. Conze; Y. Guivarc'H Limit sets of groups of linear transformations (Sankhya Series A) Tome 62 (2000), pp. 367-385 | Zbl 01644967

[6] D. Dolgopyat Prevalence of rapid mixing in hyperbolic flows, Ergodic theory and dynamical systems, Tome 18 (1998), pp. 1097-1114 | Article | MR 1653299 | Zbl 0918.58058

[7] A. Eskin; C. Mcmullen Mixing, counting and equidistribution in Lie groups, Duke mathematical journal, Tome 71 (1993), pp. 181-209 | Article | MR 1230290 | Zbl 0798.11025

[8] S. Helgason Differential geometry, Lie groups and symmetric spaces, Academic Press, San Diego, Pure and Applied Mathematics, Tome 80 (1978) | MR 514561 | Zbl 0451.53038

[9] A. Katsuda; T. Sunada Closed orbits in homology classes, Publications mathématiques de l'IHES, Tome 71 (1990), pp. 5-32 | Numdam | MR 1079641 | Zbl 0728.58026

[10] S. P. Lalley Renewal theorems in symbolic dynamics, with application to geodesic flows, noneuclidean tessellations and their fractal limits, Acta mathematica, Tome 163 (1989), pp. 1-55 | Article | MR 1007619 | Zbl 0701.58021

[11] G. Prasad $ℝ$ -regular elements in Zariski dense subgroups, Oxford quaterly journal of mathematics, Tome 45 (1994), pp. 541-545 | MR 1315463 | Zbl 0828.22010

[12] W. Parry; M. Pollicott Zeta functions and the periodic orbit structure for hyperbolic dynamics, Société Mathématique de France, Paris (Astérisque) (1990), p. 187-188 | Zbl 0726.58003

[13] M. Pollicott; R. Sharp Linear actions of free groups, Annales de l'institut Fourier, Tome 51 (2001), pp. 131-150 | Article | Numdam | MR 1821072 | Zbl 0967.37016

[14] M. Pollicott; R. Sharp Asymptotic expansions for closed orbits in homology classes, GeometriæDedicata, Tome 87 (2001), pp. 123-160 | MR 1866845 | Zbl 1049.37021

[15] J.-F. Quint Divergence exponentielle des sous-groupes discrets en rang supérieur, Commentarii Mathematicii Helvetici, Tome 77 (2002), pp. 563-608 | Article | MR 1933790 | Zbl 1010.22018

[17] J.-F. Quint L'indicateur de croissance des groupes de Schottky, Ergodic theory and dynamical systems, Tome 23 (2003), pp. 249-272 | MR 1971205 | Zbl 1037.22024

[18] Th. Roblin Ergodicité et équidistribution en courbure négative, Société Mathématique de France (Mémoires) Tome 95 (2003) | Numdam | Zbl 1056.37034

[18] C. Series The infinite word problem and limit sets in Fuchsian groups, Ergodic theory and dynamical systems, Tome 1 (1981), pp. 337-360 | MR 662473 | Zbl 0483.30029

[19] J. Tits Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, Journal f\" ur die reine und angewandte Mathematik, Tome 247 (1971), pp. 196-220 | MR 277536 | Zbl 0227.20015