Invariant distributions and X-ray transform for Anosov flows

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 invariant distributions in ${\cap }_{u<0}{H}^u(\mathcal{M})$ and whose kernel is made of coboundaries in ${\cup }_{s>0}{H}^s(\mathcal{M})$. 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.

30 pages, few corrections and new results (e.g. the image of $\Pi$ is dense among invariant distributions)

References in corpus (5)

• Patterson-Sullivan distributions and quantum ergodicity
• The inverse problem for the local geodesic ray transform
• Lens rigidity for manifolds with hyperbolic trapped set
• Invariant distributions, Beurling transforms and tensor tomography in higher dimensions
• Reconstruction formulas for X-ray transforms in negative curvature

Cited by in corpus (12)

• The X-ray transform for connections in negative curvature
• Lens rigidity for manifolds with hyperbolic trapped set
• Invariant distributions and the geodesic ray transform
• Geodesic stretch, pressure metric and marked length spectrum rigidity
• Invariant distributions, Beurling transforms and tensor tomography in higher dimensions
• Classical and microlocal analysis of the X-ray transform on Anosov manifolds
• On the s-injectivity of the X-ray transform on manifolds with hyperbolic trapped set
• Isometric extensions of Anosov flows via microlocal analysis
• Marked length spectrum rigidity for Anosov surfaces
• Mathematical Study of Scattering Resonances
• On the Pollicott-Ruelle resonances
• Smooth orbit equivalence rigidity for dissipative geodesic flows