Smooth rigidity of uniformly quasiconformal Anosov flows

Abstract

We classify the C ∞ volume-preserving uniformly quasiconformal Anosov flows, such that E + ⊕ E − is C ∞ and the dimensions of E + and E − are at least two. Then we deduce a classification of volume-preserving uniformly quasiconformal Anosov flows with smooth distributions.

On the rigidity of quasiconformal Anosov flows

“…In [Fa1], we have proved the following lemma based on [Pl1]. For the sake of completeness, let us recall the arguments.…”

“…In our previous paper [Fa1], we have studied the rigidity of volumepreserving Anosov flows with smooth E + ⊕ E − . In particular we have obtained the following Theorem 1.1.…”

“…In particular we have obtained the following Theorem 1.1. ([Fa1], Corollary 1) Let φ be a C ∞ volume-preserving quasiconformal Anosov diffeomorphism on a closed manifold Σ. If the dimensions of E + and E − are at least two, then up to finite covers φ is C ∞ conjugate to a hyperbolic automorphism of a torus.…”

“…Based on [Sa], we have found in [Fa1] for each quasiconformal Anosov flow φ t an unstable linearization {h +…”

Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

“…In higher dimensions, not much is known. In recent years, much progress has been made (see [L02,KS03,L04,F04,S05,KS07]) in the case when the stable and unstable foliations carry invariant conformal structures. To ensure existence of these conformal structures one has to at least assume that every periodic orbit has only one positive and one negative Lyapunov exponent.…”