\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points

  • * Corresponding author: Fang Wang

    * Corresponding author: Fang Wang

F. Liu is partially supported by Natural Science Foundation of Shandong Province under Grant No. ZR2020MA017, and NSFC under Grant Nos. 11301305, 11571207. F. Wang is partially supported by NSFC under Grant No. 11871045 and the State Scholarship Fund from China Scholarship Council (CSC). The research is also partially supported by key research project of the Academy for Multidisciplinary Studies, Capital Normal University

Abstract Full Text(HTML) Related Papers Cited by
  • If $ (M,g) $ is a smooth compact rank $ 1 $ Riemannian manifold without focal points, it is shown that the measure $ \mu_{\max} $ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $ \mu_{\max} $ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $ SM $ with respect to $ \mu_{\max} $ is Bernoulli is acquired provided $ M $ is a compact surface with genus greater than one and no focal points.

    Mathematics Subject Classification: Primary: 37A25, 37D40; Secondary: 53C22.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76.  doi: 10.1007/BF02773153.
    [2] W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.  doi: 10.1007/BF01456836.
    [3] W. BallmannM. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math., 122 (1985), 171-203.  doi: 10.2307/1971373.
    [4] K. Burns and A. Katok, Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317.  doi: 10.1017/S0143385700002935.
    [5] E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems, Math. USSR Izv., 5 (1971), 337-378.  doi: 10.1070/IM1971v005n02ABEH001050.
    [6] P. B. EberleinGeometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. 
    [7] R. Gulliver, On the variety of manifolds without conjugate points, Trans. Amer. Math. Soc., 210 (1975), 185-201.  doi: 10.1090/S0002-9947-1975-0383294-0.
    [8] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math., 148 (1998), 291-314.  doi: 10.2307/120995.
    [9] G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, 2002,453–545. doi: 10.1016/S1874-575X(02)80008-X.
    [10] F. LedrappierY. Lima and O. Sarig, Ergodic properties of equilibrium measures for smooth three dimensional flows, Comment. Math. Helvetici, 91 (2016), 65-106.  doi: 10.4171/CMH/378.
    [11] F. Liu and F. Wang, Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points, Acta Math. Sin. (Engl. Ser.), 32 (2016), 507-520.  doi: 10.1007/s10114-016-5200-5.
    [12] F. LiuF. Wang and W. Wu, On the Patterson-Sullivan measure for geodesic flows on rank $1$ manifolds without focal points, Discrete Contin. Dyn. Syst., 40 (2020), 1517-1554.  doi: 10.3934/dcds.2020085.
    [13] F. LiuF. Wang and W. Wu, The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth, Sci. China Math., 63 (2020), 1323-1338.  doi: 10.1007/s11425-018-9408-8.
    [14] F. Liu and X. Zhu, The transitivity of geodesic flows on rank $1$ manifolds without focal points, Differential Geom. Appl., 60 (2018), 49-53.  doi: 10.1016/j.difgeo.2018.05.007.
    [15] J. J. O'Sullivan, Riemannian manifolds without focal points, J. Differential Geometry, 11 (1976), 321-333.  doi: 10.4310/jdg/1214433590.
    [16] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.
    [17] R. O. Ruggiero, Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.  doi: 10.1017/S0143385797060963.
    [18] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
    [19] J. Watkins, The higher rank rigidity theorem for manifolds with no focal points, Geom. Dedicata, 164 (2013), 319-349.  doi: 10.1007/s10711-012-9776-3.
    [20] W. Wu, F. Liu and F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, preprint, arXiv:1812.04409.
  • 加载中
SHARE

Article Metrics

HTML views(411) PDF downloads(214) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return