Marked boundary rigidity for surfaces of Anosov type

Understand

Let $\Sigma$ be a smooth compact connected oriented surface with boundary.

• A 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 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 [arXiv:math/0305280].