Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. When $M$ is compact Francois Ledrappier showed a bijective correspondence between Hoelder cocycles $c: \Gamma \times \partial \Gamma \to \mathbb{R}$ and zero pressure Hoelder continuous functions $F:T^{1}M\to \mathbb{R}$ on the unit tangent bundle. (Page 105 of Structure au bord des variétés à courbure négative, Séminaire de théorie spectrale et géométrie, Grenoble 1994–1995, (1995), 97–122.)
The correspondence assigns each $F$ the associated weighted Busemann cocycle.
My question: is there such a correspondence if instead of being compact $M$ is assumed only to have finite Gibbs measure? How about if in addition $M$ is assumed to have finite volume? One can of course still assigns each $F$ the associated weighted Busemann cocycle but it is far from clear to me that all Hoelder cocycles are obtained this way.
