Topologie du feuilletage fortement stable
Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 981-993.

Soient X une variété de Hadamard de courbure -1 et Γ un groupe d’isométries non élémentaire. Nous montrons qu’il y a équivalence entre la non-arithméticité du spectre des longueurs de ΓX, le mélange topologique du flot géodésique et l’existence d’une feuille dense pour le feuilletage fortement stable.

Let X be a Hadamard manifold with curvature -1 and Γ be a non elementary isometry group acting freely properly discontinuously on X. We are interested in the topology of the leaves of the strong stable foliation on T 1 (ΓX). We establish equivalences between the non arithmeticity of Γ (i.e. the group generated by the length spectrum of ΓX is dense in ), the existence of a dense leaf in the non wandering set Ω X of and the topological mixing of the geodesic flow on its non wandering set. Our proof uses the action of Γ on X() and the relation between cross-ratio and length spectrum.In the case when Γ is not arithmetic, we prove that Γ is geometrically finite if and only if leaves in Ω X are dense or are associated to bounded parabolic fixed points (such leaves are closed).

@article{AIF_2000__50_3_981_0,
     author = {Dal'bo, Fran\c{c}oise},
     title = {Topologie du feuilletage fortement stable},
     journal = {Annales de l'Institut Fourier},
     pages = {981--993},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     doi = {10.5802/aif.1781},
     zbl = {0965.53054},
     mrnumber = {2001i:37045},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1781/}
}
TY  - JOUR
AU  - Dal'bo, Françoise
TI  - Topologie du feuilletage fortement stable
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 981
EP  - 993
VL  - 50
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1781/
DO  - 10.5802/aif.1781
LA  - fr
ID  - AIF_2000__50_3_981_0
ER  - 
%0 Journal Article
%A Dal'bo, Françoise
%T Topologie du feuilletage fortement stable
%J Annales de l'Institut Fourier
%D 2000
%P 981-993
%V 50
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1781/
%R 10.5802/aif.1781
%G fr
%F AIF_2000__50_3_981_0
Dal'bo, Françoise. Topologie du feuilletage fortement stable. Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 981-993. doi : 10.5802/aif.1781. https://aif.centre-mersenne.org/articles/10.5802/aif.1781/

[B] M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, L'Ens. Math., 41 (1995), 63-102. | MR | Zbl

[Bo1] B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. Jour., Vol. 77, n° 1 (1995), 229-274. | MR | Zbl

[Bo2] B. Bowditch, Relatively hyperbolic groups, Preprint 1999.

[D] F. Dal'Bo, Remarques sur le spectre des longueurs d'une surface et comptages, Bol. Soc. Bras. Math., Vol. 30, n° 2 (1999). | Zbl

[DP] F. Dal'Bo, M. Peigné, Some negatively curved manifolds with cusps, mixing and counting, J. reine angew Math., 497 (1998), 141-169. | MR | Zbl

[DS] F. Dal'Bo, A. Starkov, On a classification of limit points of infinitely generated Schottky groups, Prépublication Rennes, 1999. | Zbl

[E1] P. Eberlein, Geodesic flows on negatively curved manifolds, I, Ann. of Math., Vol. 95, n° 3 (1973), 492-510. | MR | Zbl

[E2] P. Eberlein, Geodesic flows on negatively curved manifolds, II, Trans. of the A.M.S., Vol. 178 (1973), 57-82. | MR | Zbl

[E3] P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, 1996. | MR | Zbl

[GR] Y. Guivarc'H - A. Raugi, Products of random matrices : convergence theorem, Contemp. Math., Vol. 50 (1986), 31-53. | MR | Zbl

[H] G.A. Hedlund, Fuchsian group and transitive horocycles, Duke Math. J., 2 (1936), 530-542. | JFM | Zbl

[K] I. Kim, Rigidity of rank one symmetric spaces and their product, (à paraître dans Topology). | Zbl

[NW] P. Nicholls, P. Waterman, Limit points via Schottky groups, LMS Lectures Notes, 173 (1992), 190-195. | MR | Zbl

[O] J.P. Otal, Sur la géométrie symplectique de l'espace des géodésiques d'une variété à courbure négative, Revista Mathematica Iber. Amer., 8, n° 3 (1992). | MR | Zbl

[S] M. Shub, Stabilité globale des systèmes dynamiques, Astérisque, 56 (1978). | MR | Zbl

[S1] A. Starkov, Parabolic fixed points of kleinian groups and the horospherical foliation on hyperbolic manifolds, Int. Journ. of Math., Vol. 8 n° 2 (1997), 289-299. | MR | Zbl

[S2] A. Starkov, Fuchsian groups from the dynamical viewpoint, Jour. of Dyn. and Control System, 1 (1995), 427-445. | MR | Zbl

[T] P. Tukia, Conical limit points and uniform convergence groups, J. reine angew Math., 501 (1998), 71-98. | MR | Zbl

Cité par Sources :