Invariant distributions and X-ray transform for Anosov flows

TL;DR

This paper introduces a new operator for Anosov flows that links invariant distributions with cohomological equations, and applies this to study the X-ray transform, proving injectivity on surfaces and connecting it to surjectivity.

Contribution

It defines a natural self-adjoint operator for Anosov flows, relating invariant distributions to cohomological equations, and applies it to establish properties of the X-ray transform on symmetric tensors.

Findings

The operator maps into invariant distributions and characterizes coboundaries.

Injectivity of the X-ray transform implies its surjectivity for Anosov geodesic flows.

Injectivity of the X-ray transform is proven for surfaces.

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 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.