Variation of the Liouville measure of a hyperbolic surface

Details Variation of the Liouville measure of a hyperbolic surface Variation of the Liouville measure of a hyperbolic surface

2002-03-11

arxiv.org/abs/math/0203103v2

Mathematics

Dynamical Systems

AmsLaTeX with package epsfig, 27 pages, 3 figures; one argument corrected in Section 7, minor improvements elsewhere; to appea

Scientific paper

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.

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