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Variation of the Liouville measure of a hyperbolic surface

Published online by Cambridge University Press:  20 June 2003

FRANCIS BONAHON  and
YAŞAR SÖZEN
FRANCIS BONAHON
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (e-mail: [email protected])
YAŞAR SÖZEN
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 3 , June 2003 , pp. 729 - 758
Copyright
2003 Cambridge University Press

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