INVARIANT DISTRIBUTIONS AND X-RAY TRANSFORM FOR ANOSOV FLOWS
Résumé
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 flow invariant distributions in ∩ r<0 H r (M) and whose kernel is made of coboundaries in ∪ s>0 H 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.
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