ON LOCAL AND GLOBAL RIGIDITY OF QUASI-CONFORMAL ANOSOV DIFFEOMORPHISMS

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism f of a compact manifold M. We prove that if the stable and unstable distributions have dimensions greater than two, then f is C-infinity conjugate to an affine Anosov automorphism of a finite factor of a torus. If the dimensions are at least two, the same conclusion holds under the additional assumption that M is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.