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. which maps into the space of flow invariant distributions in n boolean AND H-r< 0(r) (M) and whose kernel is made of coboundaries in.boolean OR H-s> 0(s) (M). We describe relations to the Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle M= SM of a compact manifold M, we apply this theory to study X-ray transform on symmetric tensors on M. In particular, we prove existence of flow invariant distributions on SM with prescribed push-forward on M and a similar version for tensors. This allows us to show injectivity of the X-ray transform on an Anosov surface: any divergence free symmetric tensor on M which integrates to 0 along all closed geodesics is zero.