Advanced Search
Article Contents
Article Contents

Counterexamples in non-positive curvature

Abstract Related Papers Cited by
  • We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
    Mathematics Subject Classification: Primary: 37B10, 37C40; Secondary: 34C28.


    \begin{equation} \\ \end{equation}
  • [1]

    D. V. Anosov, Geodesic flows on closed riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967).


    W. Ballmann, M. Brin and R. Spatzier, Structure of manifolds of nonpositive curvature. II, Ann. of Math., 122 (1985), 205-235.doi: 10.2307/1971303.


    P. Billingsley, Convergence of probability measures, "Wiley Series in Probability and Statistics: Probability and Statistics," 2nd edition, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.


    Yu. D. Burago and S. Z. Shefel, The geometry of surfaces in Euclidean spaces, "Geometry, III," Encyclopaedia Math. Sci., 48, Springer, Berlin, (1992), 1-85, 251-256.


    Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on non-compact spaces, Israel J. Math., 179 (2010), 157-172.doi: 10.1007/s11856-010-0076-z.


    P. Eberlein, Geodesic flows on negatively curved manifolds I, Ann. Math. II Ser., 95 (1972), 492-510.doi: 10.2307/1970869.


    P. Eberlein, "Geometry of Nonpositively Curved Manifolds," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449.


    J. Hadamard, Les surfaces courbures opposées et leurs lignes géodésiques, dans Oeuvres (1898), 2, 729-775, Paris: Editions du Centre National de la Recherche Scientifique, (1968), 2296.


    G. Knieper, Hyperbolic dynamics and Riemannian geometry, Handbook of Dynamical Systems, 1A (2002), 453-545.


    G. Link, M. Peigné and J. C. Picaud, Sur les surfaces non-compactes de rang un, L'enseignement Mathématique, 52 (2006), 3-36.


    C. Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos, "Studies in Advanced Mathematics," 2nd edition, CRC Press, Boca Raton, FL, 1999.


    K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37.doi: 10.2307/2373591.

  • 加载中

Article Metrics

HTML views() PDF downloads(69) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint