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

Yong Fang

doi:10.3934/dcds.2014.34.3471

Article Contents

2014, Volume 34, Issue 9: 3471-3483. Doi: 10.3934/dcds.2014.34.3471

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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: April 30, 2013

Revised: October 31, 2013

Published: August 2014

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Abstract

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:

Anosov flow,
entropy,
quasisymmetry.

Mathematics Subject Classification: Primary: 37A35, 34D20; Secondary: 37D35, 37D40.

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