Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties

We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the "Hamilton-Jacobi equationr" (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the $SL(2, R)$ group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time ("up to logarithms").