On uniformly quasiconformal Anosov systems

We show that for any uniformly quasiconformal symplectic Anosov diffeomorphism of a compact manifold of dimension at least 4, its finite cover is C^∞ conjugate to an Anosov automorphism of a torus. We also prove that any uniformly quasiconformal contact Anosov flow on a compact manifold of dimension at least 5 is essentially C^∞ conjugate to the geodesic flow of a manifold of constant negative curvature.