Topologie du feuilletage fortement stable

Dal'bo, Françoise

doi:10.5802/aif.1781

Dal'bo, Françoise

Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 981-993.

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).

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.1781

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     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},
     mrnumber = {2001i:37045},
     zbl = {0965.53054},
     language = {fr},
     url = {https://www.numdam.org/articles/10.5802/aif.1781/}
}

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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://www.numdam.org/articles/10.5802/aif.1781/

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