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February 2017 Invariant distributions and X-ray transform for Anosov flows
Colin Guillarmou
J. Differential Geom. 105(2): 177-208 (February 2017). DOI: 10.4310/jdg/1486522813

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 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.

Citation

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Colin Guillarmou. "Invariant distributions and X-ray transform for Anosov flows." J. Differential Geom. 105 (2) 177 - 208, February 2017. https://doi.org/10.4310/jdg/1486522813

Information

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

zbMATH: 1372.37059
MathSciNet: MR3606728
Digital Object Identifier: 10.4310/jdg/1486522813

Rights: Copyright © 2017 Lehigh University

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Vol.105 • No. 2 • February 2017
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