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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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