Gibbs u-states for the foliated geodesic flow and transverse invariant measures

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation with negatively curved leaves. On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when this foliated geodesic flow preserves a Gibbs su-state, i.e. a measure with Lebesgue disintegration both in the stable and unstable horospheres, then it has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. On the other hand we study in detail the projections of Gibbs u-states along the unit spheres tangent to the foliation. We show that they have Lebesgue disintegration in the leaves and that the local densities possess an integral representation analogue to the Poisson representation of harmonic functions.