Horocycle integral and transfer operators

A well-known example of a contact Anosov flow is the geodesic flow on the unit tangent bundle of closed Riemannian manifolds with variable negative sectional curvature. On by the Ansosov flow induced stable submanifolds one defines the horocycle flow. This horocycle flow is uniquely ergodic. What is the speed of convergence to the Birkhoff average? In a paper of Livio Flaminio and Giovanni Forni (2003) they investigated this question for the geodesic flow in the case of constant negative curvature. They found that the speed of convergence is controlled by the existence of distributions which are invariant for the adjoint of the horocycle flow. The speed of convergence is then determined by a (fractional) power spectrum with exponents associated to those eigenvalues. I report on my study of this phenomenon for contact Anosov flows of sufficient regularity. This talk serves as a rehearsal for my PhD defence. Every usueful comment is very appreciated.