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

We consider one parameter analytic hamiltonian perturbations of thegeodesic flows on surfaces of constant negative curvature. We find twodifferent necessary and sufficient conditions for the canonical equivalenceof the perturbed flows and the non-perturbed ones. One condition says thatthe "Hamilton-Jacobi equationr" (introduced in this work) for theconjugation problem should admit a solution as a formal power series (notnecessarily convergent) in the perturbation parameter. The alternativecondition is based on the identification of a complete set of invariantsfor the canonical conjugation problem. The relation with the similarproblems arising in the KAM theory of the perturbations of quasi periodichamiltonian motions is briefly discussed. As a byproduct of our analysis weobtain some results on the Livscic, Guillemin, Kazhdan equation and on theFourier series for the $SL(2, R)$ group. We also prove that the analyticfunctions on the phase space for the geodesic flow of unit speed have amixing property (with respect to the geodesic flow and to the invariantvolume measure) which is exponential with a universal exponent, independenton the particular function. This result is contrasted with the slow mixingrates that the same functions show under the horocyclic flow: in this casewe find that the decay rate is the inverse of the time ("up tologarithms")