Journal of Differential Geometry

Invariant distributions and X-ray transform for Anosov flows

Colin Guillarmou

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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 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.

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J. Differential Geom., Volume 105, Number 2 (2017), 177-208.

Received: 1 December 2014
First available in Project Euclid: 8 February 2017

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Guillarmou, Colin. Invariant distributions and X-ray transform for Anosov flows. J. Differential Geom. 105 (2017), no. 2, 177--208. doi:10.4310/jdg/1486522813.

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