  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 $f$.
