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On the mixing property for hyperbolic systems. (English) Zbl 1053.37001

The author studies mixing properties for dynamical systems with some hyperbolic behavior. As an application she proves mixing of product measures on negatively curved manifolds.
Another application is that the measure of maximal entropy on a compact manifold of nonpositive curvature is mixing, provided that the geometric rank is equal to one.

MSC:

37A25 Ergodicity, mixing, rates of mixing
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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[1] Arnold, V.; Avez, A., Ergodic Problems of Classical Mechanics (1988), Amsterdam: Addison-Wesley, Amsterdam
[2] [An] D. Anosov,Geodesic flows on Riemannian manifolds with negative curvature, Proceeings of the Steklov Institute of Mathematics90 (1967).
[3] Ballmann, W., Axial isometries of manifolds of non-positive curvature, Mathematische Annalen, 259, 131-144 (1982) · Zbl 0487.53039
[4] Ballmann, W., Lectures on spaces of nonpositive curvature (1995), Basel: Birkhäuser, Basel · Zbl 0834.53003
[5] Burns, K.; Katok, R., Manifolds with non positive curvature, Ergodic Theory and Dynamical Systems, 5, 307-317 (1985) · Zbl 0572.58019
[6] Burns, K.; Spatzier, R., Manifolds of non-positive curvature and their buildings, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 65, 35-59 (1987) · Zbl 0643.53037
[7] Dal’bo, F., Topologie du feuilletage fortement stable, Annales de l’Institut Fourier (Grenoble), 50, 981-993 (2000) · Zbl 0965.53054
[8] Dal’bo, F.; Otal, J.-P.; Peigné, M., Séries de Poincaré des groupes géométriquement finis, Israel Journal of Mathematics, 118, 109-124 (2000) · Zbl 0968.53023
[9] [E] P. Eberlein,Geometry of Non-positively Curved Manifolds, The University of Chicago Press, 1996.
[10] Eskin, A.; McMullen, C., Mixing, counting and equidistribution in Lie groups, Duke Mathematical Journal, 71, 181-209 (1993) · Zbl 0798.11025
[11] Hamenstädt, U., A new description of the Bowen-Margulis measure, Ergodic Theory and Dynamical Systems, 9, 455-464 (1989) · Zbl 0722.58029
[12] [HMP] B. Host, J.-F. Mela and F. Parreau,Analyse harmonique des mesures, Asterisque135-136 (1986). · Zbl 0589.43001
[13] Hersonsky, S.; Paulin, F., On the rigidity of discrete isometry groups of negatively curved spaces, Commentarii Mathematici Helvetici, 72, 349-388 (1997) · Zbl 0908.57009
[14] Kaimanovich, V., Invariant measures and measures at infinity, Annales de l’Institut Henri Poincaré. Physique Théorique, 53, 361-393 (1990) · Zbl 0725.58026
[15] Kaimanovich, V., Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, Journal für die reine und angewandte Mathematik, 455, 57-103 (1994) · Zbl 0803.58032
[16] Knieper, G., The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Annals of Mathematics, 148, 291-314 (1998) · Zbl 0946.53045
[17] Ledrappier, F., A renewal theorem for the distance in negative curvature, Proceedings of Symposia in Pure Mathematics, 57, 351-360 (1995) · Zbl 0842.60080
[18] Margulis, G. A., Applications of ergodic theory for the investigation of manifolds of negative curvature, Functional Analysis and its Applications, 3, 335-336 (1969) · Zbl 0207.20305
[19] [N] F. Newberger,Ergodic theory of the Bowen-Margulis measure on exotic hyperbolic spaces, Preprint, University of Maryland, 1998.
[20] Otal, J.-P., Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Revista Matemática Iberoamericana, 8, 441-456 (1992) · Zbl 0777.53042
[21] Riesz, F.; Nagy, B. Sz., Leçons d’analyse fonctionelle (1968), Paris: Gauthier-Villars, Paris
[22] [Ro] T. Roblin,Sur la théorie ergodique des groupes discrets, Thèse de l’Université d’Orsay, 1999.
[23] Rudolph, D. J., Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold, Ergodic Theory and Dynamical Systems, 2, 491-512 (1982) · Zbl 0525.58025
[24] Sarnak, P., Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Communications on Pure and Applied Mathematics, 34, 719-739 (1981) · Zbl 0501.58027
[25] Sullivan, D., Entropy, Hausdorff measures old and new and limit sets of geometrically finite Kleinian groups, Acta Mathematica, 153, 259-277 (1984) · Zbl 0566.58022
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