Variation of the Liouville Measure of a Hyperbolic Surface

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 Hölder 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 Hölder distributions for the geodesic foliation. One of the very basic examples of measure preserving dynamical system is the geodesic flow of a riemannian manifold S. The metric m of S has a natural lift to a riemannian metric m on the unit tangent bundle T S, and Liouville observed that the volume form defined by m on T S is invariant under the geodesic flow. We want to analyze what happens in this situation as we vary the metric m. In general, as we modify the metric m, the topology of the geodesic flow can dramatically change. However, if S is compact and if m has negative curvature, a fortunate phenomenon occurs: the geodesic foliation of T S, whose leaves are the orbits of the geodesic flow, is independent of the metric m. More precisely, if m and m are two negatively curved riemannian metrics on the compact manifold S, and if F and F ′ are the corresponding geodesic foliations of T S, there exists a Hölder bicontinuous homeomorphism of T S which sends F to F ′ and which is isotopic to the identity; see for instance [Gr, §8.3]. In general, this homeomorphism does not respect the parametrization of the leaves of F and F ′ provided by the geodesic flows [Ot]. As we move from the geodesic flow to the geodesic foliation, there is a one-to-one correspondence between measures on T S which are invariant under the geodesic flow and transverse measures for the geodesic foliation F . Indeed, any measure which is invariant under the geodesic flow locally is the product of some transverse measure for F and of the measure induced by the flow on the leaves of F . In particular, the Liouville measure of the metric m defines a transverse measure Lm for the geodesic foliation F of m. As indicated above, the geodesic foliation F does not depend on the negatively curved metric m; however, as a transverse measure for this fixed foliation, the Liouville transverse measure Lm does depend on m. Date: February 1, 2008. 1991 Mathematics Subject Classification. 32G15.