  We show that, on an oriented compact surface, two sufficiently $C^2$-close
Riemannian metrics with strictly convex boundary, no conjugate points,
hyperbolic trapped set for their geodesic flows, and same marked boundary
distance, are isometric via a diffeomorphism that fixes the boundary. We also
prove that the same conclusion holds on a compact surface for any two
negatively curved Riemannian metrics with strictly convex boundary and same
marked boundary distance, extending a result of Croke and Otal.
