The invariant connection of a 1/2-pinched Anosov diffeomorphism and rigidity

Abstract

Let / be a C°° Anosov diffeomorphism of a compact manifold M, preserving a smooth measure. If / satisfies the |pinching assumption defined below, it must preserve a continuous affine connection for which the leaves of the Anosov foliations are totally geodesic, geodesically complete, and flat (its tangential curvature is defined along individual leaves). If this connection, which is the unique /-invariant affine connection on M, is Cr-differentiable, r > 2, then / is conjugate via a Cr+2-affine diffeomorphism to a hyperbolic automorphism of a geodesically complete flat manifold. If / preserves a smooth symplectic form, has C3 Anosov foliations, and satisfies the 2 : 1-nonresonance condition (an assumption that is weaker than pinching), then / is C°° conjugate to a hyperbolic automorphism of a complete flat manifold. (In the symplectic case, the invariant connection is the one previously defined by Kanai in the context of geodesic flows.) If the foliations are C2 and the holonomy pseudo-groups satisfy a certain growth conditipn, the same conclusion holds.