Topologie du feuilletage fortement stable

Dal'bo, Françoise

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

Soient $X$ une variété de Hadamard de courbure $\le -1$ et $\Gamma$ un groupe d’isométries non élémentaire. Nous montrons qu’il y a équivalence entre la non-arithméticité du spectre des longueurs de $\Gamma \setminus X$, 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 $\le -1$ and $\Gamma$ 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 $T^1(\Gamma \setminus X)$. We establish equivalences between the non arithmeticity of $\Gamma$ (i.e. the group generated by the length spectrum of $\Gamma \setminus X$ is dense in $ℝ$), the existence of a dense leaf in the non wandering set ${\Omega }^X$ of $ℱ$ and the topological mixing of the geodesic flow on its non wandering set. Our proof uses the action of $\Gamma$ on $X\left(\infty \right)$ and the relation between cross-ratio and length spectrum.In the case when $\Gamma$ is not arithmetic, we prove that $\Gamma$ is geometrically finite if and only if leaves in ${\Omega }^X$ are dense or are associated to bounded parabolic fixed points (such leaves are closed).

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