2013/11/27 by Sébastien Alvarez, Alvarez, Sébastien
Mathematics · #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1311.7121
This paper is devoted to the study of Gibbs u-states for the geodesic flow\ntangent to a foliation with negatively curved leaves. On the one hand we give\nsufficient conditions for the existence of transverse invariant measures. In\nparticular we prove that when this foliated geodesic flow preserves a Gibbs\nsu-state, i.e. a measure with Lebesgue disintegration both in the stable and\nunstable horospheres, then it has to be obtained by combining a transverse\ninvariant measure and the Liouville measure on the leaves. On the other hand we\nstudy in detail the projections of Gibbs u-states along the unit spheres\ntangent to the foliation. We show that they have Lebesgue disintegration in the\nleaves and that the local densities possess an integral representation analogue\nto the Poisson representation of harmonic functions.\n