We consider a transitive uniformly quasi-conformal Anosov diffeomorphism $f$ of a compact manifold $\mathcal{M}$. We prove that if the stable and unstable distributions have dimensions greater than two, then $f$ is $C^\infty$ 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 $\mathcal{M}$ is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.

AMS 2000 Mathematics subject classification: Primary 37C; 37D