INVARIANT DISTRIBUTIONS AND X-RAY TRANSFORM FOR ANOSOV FLOWS

For Anosov flows preserving a smooth measure on a closed manifold M, we define a natural self-adjoint operator Π which maps into the space of in-variant distributions in ∩ u<0 H u (M) and whose kernel is made of coboundaries in ∪ s>0 H s (M). We describe relations to Livsic theorem and recover regularity proper-ties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle M = SM of a compact manifold, we apply this theory to study questions related to X-ray transform on symmetric tensors on M : in particular we prove that injectivity implies surjectivity of X-ray transform, and we show injectivity for surfaces.