Burns–Katok marked length spectrum rigidity conjecture

Let $M$ be a smooth $n$-dimensional closed manifold such that the space of Anosov metrics $\mathcal{M}_{\mathrm{Anosov}}(M)$ is nonempty. The marked length spectrum assigns to each isometry class its marked length function, giving a map

$$L: \mathbb{M}_{\mathrm{Anosov}} \to \ell^\infty(\mathcal{C}).$$

Burns–Katok conjecture. The marked length spectrum map $L$ is injective.

This conjecture asserts that the marked lengths of closed geodesics determine an Anosov metric up to an isometry isotopic to the identity. It is a central rigidity problem, known in important special cases but open in the stated generality.