Spectral rigidity and invariant distributions on Anosov surfaces. (English) Zbl 1304.37021
Authors’ abstract: This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface \((M, g)\), given a smooth function \(f\) on \(M\), there is a distribution in the Sobolev space \(H^{-1}(SM)\) that is invariant under the geodesic flow and whose projection to \(M\) is the given function \(f\).
Reviewer: Kazuhiro Sakai (Utsunomiya)
MSC:
37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
53C22 | Geodesics in global differential geometry |
53C65 | Integral geometry |
53D25 | Geodesic flows in symplectic geometry and contact geometry |