Symplectic rigidity for Anosov hypersurfaces

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.