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 \(T^1(M)\).


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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[1] [B] , Structure conforme au bord et flot géodésique d’un CAT(-1)-espace, L’Ens. Math., 41 (1995), 63-102. · Zbl 0871.58069
[2] [Bo1] , Geometrical finiteness with variable negative curvature, Duke Math. Jour., Vol. 77, n° 1 (1995), 229-274. · Zbl 0877.57018
[3] [Bo2] , Relatively hyperbolic groups, Preprint 1999. · Zbl 1259.20052
[4] [D] , Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Bras. Math., Vol. 30, n° 2 (1999). · Zbl 1058.53063
[5] [DP] , , Some negatively curved manifolds with cusps, mixing and counting, J. reine angew Math., 497 (1998), 141-169. · Zbl 0890.53043
[6] [DS] , , On a classification of limit points of infinitely generated Schottky groups, Prépublication Rennes, 1999. · Zbl 0963.30028
[7] [E1] , Geodesic flows on negatively curved manifolds, I, Ann. of Math., Vol. 95, n° 3 (1973), 492-510. · Zbl 0217.47304
[8] [E2] , Geodesic flows on negatively curved manifolds, II, Trans. of the A.M.S., Vol. 178 (1973), 57-82. · Zbl 0264.53027
[9] [E3] , Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, 1996. · Zbl 0883.53003
[10] [GR] - , Products of random matrices : convergence theorem, Contemp. Math., Vol. 50 (1986), 31-53. · Zbl 0592.60015
[11] [H] , Fuchsian group and transitive horocycles, Duke Math. J., 2 (1936), 530-542. · JFM 62.0392.03
[12] [K] , Rigidity of rank one symmetric spaces and their product, (à paraître dans Topology). · Zbl 0997.53034
[13] [NW] , , Limit points via Schottky groups, LMS Lectures Notes, 173 (1992), 190-195. · Zbl 0767.30034
[14] [O] , Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Revista Mathematica Iber. Amer., 8, n° 3 (1992). · Zbl 0777.53042
[15] [S] , Stabilité globale des systèmes dynamiques, Astérisque, 56 (1978). · Zbl 0396.58014
[16] [S1] , Parabolic fixed points of kleinian groups and the horospherical foliation on hyperbolic manifolds, Int. Journ. of Math., Vol. 8 n° 2 (1997), 289-299. · Zbl 0874.57027
[17] [AHS78] , and , Schrödinger operators with magnetic fields, I. General interactions, Duke Math. J., 45 (1978 · Zbl 0986.37033
[18] [T] , Conical limit points and uniform convergence groups, J. reine angew Math., 501 (1998), 71-98. · Zbl 0909.30034
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