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Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances
1. | Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex |
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.
References:
[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. |
show all references
References:
[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. |
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