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September  2014, 34(9): 3471-3483. doi: 10.3934/dcds.2014.34.3471

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

Yong Fang 1,

1. 

Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex

Received  May 2013 Revised  November 2013 Published  March 2014

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.
Keywords: entropy, Anosov flow, quasisymmetry..
Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D4.
Citation: Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471
References:
[1]

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

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

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

[9]

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

[10]

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

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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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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