On the rigidity of quasiconformal Anosov flows

Abstract

We develop further our study of quasiconformal Anosov flows in our previous (Y. Fang. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys.24 (2004), 1–23). For example, we prove the following result: Let φ be a transversely symplectic Anosov flow with dim Ess≥2 and dim Esu≥2. If φ is quasiconformal, then it is, up to finite covers, $C^\infty $ orbit equivalent either to the suspension of a symplectic hyperbolic automorphism of a torus or to the geodesic flow of a closed hyperbolic manifold.

Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows

“…In addition since the dimension of M is supposed to be at least three throughout this paper, then by the celebrated rigidity theorem of Mostow (see [18]) g is up to a constant scale and finite covers isometric to the initial hyperbolic Riemannian metric g. Thus up to a constant change of time scale, ψ λ is C 2 flow equivalent to the geodesic flow of g. We deduce that the Euler-Lagrange flow ϕ λ is C 2 orbit equivalent to the geodesic flow of g. In [6] we proved the following rigidity result, which will complete the proof of Theorem 1.4: Theorem 5.1 Let ϕ be the geodesic flow of a real hyperbolic Riemannian manifold of dimension at least three, and ψ be a C ∞ Anosov flow. If ψ is C 1 orbit equivalent to ϕ, then these two flows must be C ∞ orbit equivalent.…”

Stability of Anosov Hamiltonian Structures

“…Remark 5.8. Actually for the case n = 3 we could bypass the above and conclude immediately that φ 1 t is C ∞ -orbit equivalent to the geodesic flow of a closed threedimensional hyperbolic manifold using recent work of Fang [12]. Namely, for n = 3 (so dim E u = 2), an almost complex structure is the 'same' as having a conformal structure, and thus the fact that E u admits a φ 1 t -invariant almost complex structure J implies that φ 1 t is quasiconformal (see [12]).…”

“…Actually for the case n = 3 we could bypass the above and conclude immediately that φ 1 t is C ∞ -orbit equivalent to the geodesic flow of a closed threedimensional hyperbolic manifold using recent work of Fang [12]. Namely, for n = 3 (so dim E u = 2), an almost complex structure is the 'same' as having a conformal structure, and thus the fact that E u admits a φ 1 t -invariant almost complex structure J implies that φ 1 t is quasiconformal (see [12]). Then [12,Theorem 3] tells us that up to finite covers, φ 1 t is C ∞ -orbit equivalent to the suspension of a symplectic hyperbolic automorphism or to the geodesic flow of a closed three-dimensional hyperbolic manifold.…”

“…Namely, for n = 3 (so dim E u = 2), an almost complex structure is the 'same' as having a conformal structure, and thus the fact that E u admits a φ 1 t -invariant almost complex structure J implies that φ 1 t is quasiconformal (see [12]). Then [12,Theorem 3] tells us that up to finite covers, φ 1 t is C ∞ -orbit equivalent to the suspension of a symplectic hyperbolic automorphism or to the geodesic flow of a closed three-dimensional hyperbolic manifold. The former is impossible by Theorem 4.2.…”

Normal forms for non-uniform contractions

“…More precisely, we gave a construction of H x that depend smoothly on x along the leaves and proved that they define an atlas with transition maps in a finite dimensional Lie group. Non-stationary normal forms were used extensively in the study of rigidity of uniformly hyperbolic dynamical systems and group actions, see for example [KtSp97,KS03,KS06,F07,FFH10,GoKS11,FKSp11].…”