Let \(\Sigma \) be a smooth compact connected oriented surface with non-empty boundary. A Riemannian metric on \(\Sigma \) is said to be of Anosov type if it has strictly convex boundary, no conjugate points, and a hyperbolic trapped set. We prove that two Riemannian metrics of Anosov type with the same marked boundary distance are isometric (via a boundary-preserving isometry isotopic to the identity). As a corollary, we retrieve the boundary distance rigidity result for simple disks of Pestov and Uhlmann (Ann Math (2) 161(2):1093–1110, 2005). The proof rests on a new transfer principle showing that, in any dimension, the marked length spectrum rigidity conjecture implies the marked boundary distance rigidity conjecture under the existence of a suitable isometric embedding into a closed Anosov Riemannian manifold. Such an isometric embedding result for open Riemannian surfaces of Anosov type was proved by the first author with Chen and Gogolev (Journal de l’École polytechnique-Math. Tome 10:945–987, 2023) while the marked length spectrum rigidity for closed Anosov Riemannian surfaces was established by the second author with Guillarmou and Paternain (Marked length spectrum rigidity for Anosov surfaces. arXiv e-prints, arXiv:2303.12007, 2023).
