# 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 & Continuous Dynamical Systems, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471
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