Spectral rigidity and invariant distributions on Anosov surfaces
Paternain, G., Salo, M., & Uhlmann, G. (2014). Spectral rigidity and invariant distributions on Anosov surfaces. Journal of differential geometry, 98 (1), 147-181. Retrieved from http://projecteuclid.org/euclid.jdg/1406137697
Published inJournal of Differential Geometry
© The Authors. © Lehigh University, 2014. This is an authors' final draft version of an article whose final and definitive form has been published by Lehigh University. Published in this repository with the kind permission of the publisher.
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.