# American Institute of Mathematical Sciences

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

 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.
Citation: Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete and 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. [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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##### 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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