Invariant distributions and X-ray transform for Anosov flows

Abstract

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 flow invariant distributions in $\cap_{r \lt 0} H^r (\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s \gt 0} H^s (\mathcal{M})$. 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 $\mathcal{M}= SM$ of a compact manifold $\mathcal{M}$, we apply this theory to study X-ray transform on symmetric tensors on $\mathcal{M}$. In particular, we prove existence of flow invariant distributions on $SM$ with prescribed push-forward on $\mathcal{M}$ 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 $\mathcal{M}$ which integrates to $0$ along all closed geodesics is zero.