no. 3

Orbital counting for some convergent groups  [ Comptage orbital pour certains groupes convergents ]
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1307-1340.

Nous construisons des variétés géométriquement finies à courbure strictement négative pincée, dont le flot géodésique possède une mesure de Bowen-Margulis non ergodique infinie, et dont la série de Poincaré converge à l’exposant δ Γ , et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour tout α∈]1,2[ et toute fonction à variations lentes L:ℝ→(0,+∞), nous construisons des variétés de Hadamard (X,g) de dimension N≥2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsque R→+∞,

NΓ(R):=♯{γ∈Γ|d(o,γ·o)≤R}∼CΓ(o)L(R)RαeδΓR,

où C Γ (o) est une constante strictement positive qui dépend du point o.

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen–Margulis measure and whose Poincaré series converges at the critical exponent δ Γ . We obtain an explicit asymptotic for their orbital growth function. Namely, for any α∈]1,2[ and any smooth slowly varying function L:ℝ→(0,+∞), we construct N-dimensional Hadamard manifolds (X,g) of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups Γ such that, as R→+∞,

NΓ(R):=♯{γ∈Γ|d(o,γ·o)≤R}∼CΓ(o)L(R)RαeδΓR,

for some C Γ (o)>0 depending on the base point o.

Reçu le : 2018-10-01
Révisé le : 2019-06-13
Accepté le : 2019-07-11
Publié le : 2020-12-18
DOI : https://doi.org/10.5802/aif.3335
Classification : 58F17,  58F20,  20H10
Mots clés : exposant de Poincaré, groupe convergent/divergent, fonction orbitale 
@article{AIF_2020__70_3_1307_0,
     author = {Peign\'e, Marc and Tapie, Samuel and Vidotto, Pierre},
     title = {Orbital counting for some convergent groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1307--1340},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.5802/aif.3335},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1307_0/}
}
Peigné, Marc; Tapie, Samuel; Vidotto, Pierre. Orbital counting for some convergent groups. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1307-1340. doi : 10.5802/aif.3335. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1307_0/

[1] Babillot, Martine; Peigné, Marc Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps, Bull. Soc. Math. Fr., Volume 134 (2006) no. 1, pp. 119-163 | Article | MR 2233702

[2] Bourdon, Marc Structure conforme au bord et flot géodésique d’un CAT (-1)-espace, Enseign. Math., Volume 41 (1995) no. 1-2, pp. 63-102 | MR 1341941

[3] Bowditch, Brian H. Geometrical finiteness with variable negative curvature, Duke Math. J., Volume 77 (1995) no. 1, pp. 229-274 | Article | MR 1317633

[4] do Carmo, Manfredo Perdigão Riemannian geometry, Mathematics: Theory Applications, Birkhäuser, 1992, xiv+300 pages (Translated from the second Portuguese edition by Francis Flaherty) | Article | MR 1138207

[5] do Carmo, Manfredo Perdigão Differential geometry of curves & surfaces, Dover Publications, 2016, xvi+510 pages | MR 3837152

[6] Dal’bo, Françoise; Otal, Jean-Pierre; Peigné, Marc Séries de Poincaré des groupes géométriquement finis, Isr. J. Math., Volume 118 (2000), pp. 109-124 | Article | MR 1776078

[7] Dal’bo, Françoise; Peigné, Marc Groupes du ping-pong et géodésiques fermées en courbure -1, Ann. Inst. Fourier, Volume 46 (1996) no. 3, pp. 755-799 | MR 1411728

[8] Dal’bo, Françoise; Peigné, Marc; Picaud, Jean-Claude; Sambusetti, Andrea Convergence and counting in infinite measure, Ann. Inst. Fourier, Volume 67 (2017) no. 2, pp. 483-520 | MR 3669504

[9] Gouëzel, Sébastien Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math., Volume 125 (2011) no. 2, pp. 193-212 | Article | MR 2871313

[10] Hennion, Hubert Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Am. Math. Soc., Volume 118 (1993) no. 2, pp. 627-634 | Article | MR 1129880

[11] Margulis, Grigoriǐ Aleksandrovich Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkts. Anal. Prilozh., Volume 3 (1969) no. 4, p. 89-90 | MR 0257933

[12] Otal, Jean-Pierre; Peigné, Marc Principe variationnel et groupes kleiniens, Duke Math. J., Volume 125 (2004) no. 1, pp. 15-44 | Article | MR 2097356

[13] Peigné, Marc On the Patterson–Sullivan measure of some discrete group of isometries, Isr. J. Math., Volume 133 (2003), pp. 77-88 | Article | MR 1968423

[14] Peigné, Marc On some exotic Schottky groups, Discrete Contin. Dyn. Syst., Volume 31 (2011) no. 2, pp. 559-579 | Article | MR 2805820

[15] Pit, Vincent; Schapira, Barbara Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature, Ann. Inst. Fourier, Volume 68 (2018) no. 2, pp. 457-510 | MR 3803108

[16] Pollicott, Mark; Sharp, Richard Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature, Invent. Math., Volume 117 (1994) no. 2, pp. 275-302 | Article | MR 1273266

[17] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr., Nouv. Sér. (2003) no. 95, p. vi+96 | Article | MR 2057305

[18] Vidotto, Pierre Ergodic properties of some negatively curved manifolds with infinite measure, Mém. Soc. Math. Fr., Nouv. Sér. (2019) no. 160, p. vi+132 | MR 3939852