For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{ M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of invariant distributions in $\cap_{ u<0} H^{ u}(\mathcal{ M})$ and whose kernel is made of coboundaries in $\cup_{ s>0} H^{ s}(\mathcal{ M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{ M}=SM$ 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.
Categories: math.AP math.DG math.DS