Differentiability, rigidity and Godbillon-Vey classes for Anosov flows

The central geometric objects associated with an Anosov dynamical system on a compact manifold are the invariant stable and unstable foliations. While each stable and unstable manifold is as smooth as the system itself, the foliations that they form are believed to have only a moderate degree of regularity for most systems. We will analyze the exact degree of regularity of codimension-one stable and unstable foliations for low dimensional systems. Our main results relate the regularity of these foliations to cohomology classes associated to the system: the Anosov class, a new invariant of the flow which we introduce in this paper, and the Godbillon-Vey class of the weak-stable foliations, which we show is a well-defined invariant of the system