Gibbs measures for foliated bundles with negatively curved leaves

Abstract

In this paper we develop a notion of Gibbs measure for the geodesic flow tangent to a foliated bundle over a compact and negatively curved basis. We also develop a notion of $F$-harmonic measure and prove that there exists a natural bijective correspondence between the two. For projective foliated bundles with $\mathbb{C}\mathbb{P}^1$-fibers without transverse invariant measure, we show the uniqueness of these measures for any H\"older potential on the basis. In that case we also prove that $F$-harmonic measures are realized as weighted limits of large balls tangent to the leaves and that their conditional measures on the fibers are limits of weighted averages on the orbits of the holonomy group.