Rigidity of higher-dimensional conformal Anosov systems

Abstract

We show that Anosov systems in manifolds with trivial tangent bundles and with the property that the derivatives of the return maps at periodic orbits are multiples of the identity in the stable and unstable bundles are locally rigid. That is, any other smooth map, in a C^1 neighborhood such that it has the same Jordan normal form at corresponding periodic orbits is smoothly conjugate to it. This generalizes results of Castro and Moriyón (1997). We present several arguments for the main results. In particular, we use quasi-conformal regularity theory. We also extend the examples of an earlier paper of the author (1992) to show that some of the hypotheses we make in this paper are indeed necessary.