Topologie du feuilletage fortement stable

Dal'bo, Françoise

Annales de l'Institut Fourier, Tome 50 (2000), p. 981-993 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Soient $X$ une variété de Hadamard de courbure $\leq - 1$ et $Γ$ un groupe d’isométries non élémentaire. Nous montrons qu’il y a équivalence entre la non-arithméticité du spectre des longueurs de $Γ ∖ X$ , 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 $\leq - 1$ 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 $ℱ$ on $T^{1} (Γ ∖ X)$ . We establish equivalences between the non arithmeticity of $Γ$ (i.e. the group generated by the length spectrum of $Γ ∖ X$ is dense in $ℝ$ ), the existence of a dense leaf in the non wandering set $Ω^{X}$ of $ℱ$ and the topological mixing of the geodesic flow on its non wandering set. Our proof uses the action of $Γ$ on $X (\infty)$ 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).

Publié le : 2000-01-01

MR 2001i:37045 | Zbl 0965.53054 | 3 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1781

@article{AIF_2000__50_3_981_0,
     author = {Dal'bo, Fran\c coise},
     title = {Topologie du feuilletage fortement stable},
     journal = {Annales de l'Institut Fourier},
     volume = {50},
     year = {2000},
     pages = {981-993},
     doi = {10.5802/aif.1781},
     mrnumber = {2001i:37045},
     zbl = {0965.53054},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_3_981_0}
}

Dal'bo, Françoise. Topologie du feuilletage fortement stable. Annales de l'Institut Fourier, Tome 50 (2000) pp. 981-993. doi : 10.5802/aif.1781. http://gdmltest.u-ga.fr/item/AIF_2000__50_3_981_0/

Bibliographie

[B] M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, L'Ens. Math., 41 (1995), 63-102. | MR 96f:58120 | Zbl 0871.58069

[Bo1] B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. Jour., Vol. 77, n° 1 (1995), 229-274. | MR 96b:53056 | Zbl 0877.57018

[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 1058.53063

[DP] F. Dal'Bo, M. Peigné, Some negatively curved manifolds with cusps, mixing and counting, J. reine angew Math., 497 (1998), 141-169. | MR 99c:58124 | Zbl 0890.53043

[DS] F. Dal'Bo, A. Starkov, On a classification of limit points of infinitely generated Schottky groups, Prépublication Rennes, 1999. | Zbl 0963.30028

[E1] P. Eberlein, Geodesic flows on negatively curved manifolds, I, Ann. of Math., Vol. 95, n° 3 (1973), 492-510. | MR 46 #10024 | Zbl 0217.47304

[E2] P. Eberlein, Geodesic flows on negatively curved manifolds, II, Trans. of the A.M.S., Vol. 178 (1973), 57-82. | MR 47 #2636 | Zbl 0264.53027

[E3] P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, 1996. | MR 98h:53002 | Zbl 0883.53003

[GR] Y. Guivarc'H - A. Raugi, Products of random matrices : convergence theorem, Contemp. Math., Vol. 50 (1986), 31-53. | MR 87m:60024 | Zbl 0592.60015

[H] G.A. Hedlund, Fuchsian group and transitive horocycles, Duke Math. J., 2 (1936), 530-542. | JFM 62.0392.03 | Zbl 0015.10201

[K] I. Kim, Rigidity of rank one symmetric spaces and their product, (à paraître dans Topology). | Zbl 01708462

[NW] P. Nicholls, P. Waterman, Limit points via Schottky groups, LMS Lectures Notes, 173 (1992), 190-195. | MR 94a:20082 | Zbl 0767.30034

[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 94a:58077 | Zbl 0777.53042

[S] M. Shub, Stabilité globale des systèmes dynamiques, Astérisque, 56 (1978). | MR 80c:58015 | Zbl 0396.58014

[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 98d:58144 | Zbl 0874.57027

[S2] A. Starkov, Fuchsian groups from the dynamical viewpoint, Jour. of Dyn. and Control System, 1 (1995), 427-445. | MR 96i:58140 | Zbl 0986.37033

[T] P. Tukia, Conical limit points and uniform convergence groups, J. reine angew Math., 501 (1998), 71-98. | MR 2000b:30067 | Zbl 0909.30034