We consider a closed negatively curved surface (M, g) with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric g₀ on M. We show there is a smooth diffeomorphism F: M M with derivative bounds close to 1, depending on the ratio of the two marked length spectrum functions. This is a two-dimensional version of our main result in But25b.
Karen Butt (Sat,) studied this question.