  For a compact riemannian manifold of negative curvature, the geodesic
foliation of its unit tangent bundle is independent of the negatively curved
metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric
defines a natural transverse measure to this foliation, the Liouville
transverse measure, which does depend on the metric. For a surface S, we show
that the map which to a hyperbolic metric on S associates its Liouville
transverse measure is differentiable, in an appropriate sense. Its tangent map
is valued in the space of transverse Holder distributions for the geodesic
foliation.
