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

Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances

Abstract Related Papers Cited by
  • This paper deals with interactions between metric quasiconformal geometry and the rigidity of Anosov flows. In the first part of this article, we study a canonical time change of Anosov flows. Then we use it to obtain the thorough classification of volume-preserving quasiconformal Anosov flows.
        Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence:
        Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$.
        If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.
    Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D40.

    Citation:

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

    V. D. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Pros. Inst. Steklov, 90 (1967), 1-235.

    [2]

    Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74.doi: 10.2307/2152750.

    [3]

    Y. Fang, Smoth rigidity of uniformly quasiconformal Anosov flows, Ergodic theory and Dynam. Systems, 24 (2004), 1937-1959.doi: 10.1017/S0143385704000264.

    [4]

    Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, 27 (2007), 1773-1802.doi: 10.1017/S0143385707000326.

    [5]

    Y. Fang, Thermodynamic invariants of Anosov flows and rigidity, Continuous and Discrete Dynamical Systems, 24 (2009), 1185-1204.doi: 10.3934/dcds.2009.24.1185.

    [6]

    Y. Fang, Geometric Anosov flows of dimension five with smooth distributions, Journal of the Inst. of Math. Jussieu, 4 (2005), 1-30.doi: 10.1017/S1474748005000083.

    [7]

    P. Foulon, Entropy rigidity of Anosov flows in dimension three, Ergodic Theory and Dynam. Systems, 21 (2001), 1101-1112.doi: 10.1017/S0143385701001523.

    [8]

    E. Ghys, Déformation des flots d'Anosov et e groupes fuchsiens, Ann. Inst. Fourier, 42 (1992), 209-247.doi: 10.5802/aif.1290.

    [9]

    E. Ghys, Flots d'Anosov dont les feuilletages stable et instable sont différentiables, Ann. Scient. Ec. Norm. Sup., 20 (1987), 251-270.

    [10]

    U. Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory and Dynam. Systems, 9 (1989), 455-464.doi: 10.1017/S0143385700005095.

    [11]

    B. Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory and Dynam. Systems, 9 (1989), 465-468.doi: 10.1017/S0143385700005101.

    [12]

    J. Heinonen, What is a quasiconformal mapping?, Notices of the AMS, 53 (2006), 1334-1335.

    [13]

    J. Heinonen, Lectures on Analysis on metric spaces, Universitext, Springer, 2001.doi: 10.1007/978-1-4613-0131-8.

    [14]

    B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

    [15]

    M. W. Hirsch and C. C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238.

    [16]

    M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30.

    [17]

    W. Parry, Synchronization of canonical measures for hyperbolic attractors, Commun.Math. Phys., 106 (1986), 267-275.doi: 10.1007/BF01454975.

    [18]

    V. Sadovskaya, On uniformly quasiconformal Anosov flows, Math. Res. Lett., 12 (2005), 425-441.doi: 10.4310/MRL.2005.v12.n3.a12.

    [19]

    V. Sadovskaya and B. Kalinin, On local and global rigidity of quasiconformal Anosov diffeomorphisms, J. Inst. Math. Jussieu, 2 (2003), 567-582.doi: 10.1017/S1474748003000161.

    [20]

    C. Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett., 2 (1995), 327-338.doi: 10.4310/MRL.1995.v2.n3.a10.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return