In this paper we establish Livshitz-type theorems for partially hyperbolic systems. To be more precise, we prove that for a large class of partially hyperbolic transformations and flows the subspace of Hoelder coboundaries is closed and can be described by some natural geometric conditions. This class includes an open, in C2 topology, neighborhood of the time-one maps of contact Anosov flows (for example, the geodesic flows on manifolds of negative curvature). Along the way we prove several results on the transitivity of the pair of stable and unstable foliations for partially hyperbolic systems. In particular, we establish the transitivity property for the time-one maps of contact Anosov flows and their small perturbations, which has important applications to the stable ergodicity of the time-one maps of geodesic flows on the manifolds of negative curvature.
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