  We show, using standard results in length spectrum rigidity and symplectic
homology, that if the unit tangent bundles of two compact surfaces of negative
curvature are exact symplectomorphic, then the underlying surfaces are
isometric, and hence the disk bundles are symplectomorphic smoothly up to the
boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic
boundaries, we try to extend the above result to other classes of domains in
(co)tangent bundles of surfaces. The domains we can treat consist of those
bounded by deformations of negatively curved unit cotangent bundles through
boundaries with Anosov characteristic (Reeb) flows. We show that if two such
are exact symplectomorphic, then the two domains are symplectomorphic by a
diffeomorphism smooth up to the boundary. Furthermore, if two such surfaces
have the same marked length spectrum then they are connected by a 1-parameter
family of domains bounded by hypersurfaces with smoothly time preserving
conjugate Reeb flows by a smooth family of Hamiltonian diffeomorphisms. A few
counterexamples to easy generalizations are given.
