Cocycles, Hausdorff measures and cross ratios

Let $f$ be a flip-invariant Hölder continuous function on the unit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold $M$. We show that conditionals on strong unstable manifolds of the Gibbs equilibrium state defined by $f$ can be realized as Hausdorff measures. Moreover, cohomology classes of flip invariant cocycles are in one-to-one correspondence to cross ratios on the space of four pairwise distinct points of the ideal boundary of the universal covering $\tilde M$ of $M$.

Anosov foliations and cohomology

The cohomology action of an Anosov diffeomorphism on a nilmanifold resembles that of a Cartesian product map. Corresponding results hold for infranilmanifolds, giving an invariant bigrading of the cohomology and a fourfold symmetry that extends Poincaré duality. Holonomy invariant cocycles are applied to the action on first cohomology.