Invariant distributions and X-ray transform for Anosov flows

By C. Guillarmou. 2014

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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 $\cap_{u 0} H^{s}(\mathcal{M})$. 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/379b79f76c529108a638d22acbd43486f636db87