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Article Dans Une Revue Journal of Differential Geometry Année : 2017

INVARIANT DISTRIBUTIONS AND X-RAY TRANSFORM FOR ANOSOV FLOWS

Résumé

For Anosov flows preserving a smooth measure on a closed manifold M, we define a natural self-adjoint operator Π which maps into the space of flow invariant distributions in ∩ r<0 H r (M) and whose kernel is made of coboundaries in ∪ s>0 H s (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 M = SM of a compact manifold M , we apply this theory to study X-ray transform on symmetric tensors on M. In particular we prove existence of flow invariant distributions on SM with prescribed push-forward on 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 M which integrates to 0 along all closed geodesics is zero.
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Dates et versions

hal-01236227 , version 1 (01-12-2015)

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Colin Guillarmou. INVARIANT DISTRIBUTIONS AND X-RAY TRANSFORM FOR ANOSOV FLOWS. Journal of Differential Geometry, 2017, 105 (2), pp.177--208. ⟨hal-01236227⟩
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