**Authors**:
P. Collet, H. Epstein, G. Gallavotti

**Title**:
*
Perturbations of Geodesic Flows on Surfaces of Constant
Negative Curvature and Their Mixing Properties
*

Communication in Mathematical Physics, ** 95**, 61--112, 1984

**Abstract:** Abstract. 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 equation" (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, equal to the curvature of the surface divided by $2$. 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").

**Key words: **
Classical Mechanics, Integrability, Hyperbolic geometry,
Harmonic analysis, SL(2,R), Canonical congjugation, Canonical maps