## Topologie du feuilletage fortement stable. (Topology of the strong stable foliation).(French)Zbl 0965.53054

Consider a homogeneous space $$M$$ defined by the free, discontinuous, proper action of a group of isometries $$\Gamma$$ on a connected, simply connected, complete Riemannian manifold $$X$$ with sectional curvature $$K$$ normalized so that $$\sup K=-1.$$ Define a foliation on the unit tangent bundle $$T^1(M)$$ of $$M$$, in which two unit tangent vectors belong to the same leaf if the positive geodesic rays determined by corresponding unit tangent vectors to $$X$$ have the same endpoint on the boundary at infinity of $$X$$ and the horocycles in $$X$$ with centre at this common boundary point and passing through the two base points of the tangent vectors to $$X$$ are identical. The author studies conditions that the group $$\Gamma$$ is not arithmetic, and finds that this is equivalent to the restriction of the above foliation to the limit set at infinity for an orbit of $$\Gamma$$ is topologically transitive (contains a dense leaf) and also equivalent to topological mixing of the recurrent part of the geodesic flow on $$T^1(M)$$.

### MSC:

 53D25 Geodesic flows in symplectic geometry and contact geometry 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 53C12 Foliations (differential geometric aspects) 53C30 Differential geometry of homogeneous manifolds
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