A counterexample to marked length spectrum semi-rigidity

Given a closed orientable negatively curved Riemannian surface $(M,g)$, we show how to construct a perturbation $(M,g^{\mathrm{\prime }})$ such that each closed geodesic becomes longer, and yet there is no diffeomorphism $f:(M,g^{\mathrm{\prime }})\to (M,g)$ which contracts every tangent vector.