This analysis demonstrates how marked length spectrum results yield a diffeomorphism on surfaces, indicating rigidity properties.
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].
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Karen Butt (2025) studied this question.
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