Invariant distributions and the transport twistor space of closed surfaces

We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on $S^2$ are determined by their zeroth and first Fourier modes.