Quantitative marked length spectrum rigidity

We consider a closed Riemannian manifold M of negative curvature and dimension at least 3, with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space N .Using the methods of Hamenstdt (1999), we show that the volumes of M and N are approximately equal.We then show that the smooth map F W M !N constructed by Besson, Courtois and Gallot (1996) is a diffeomorphism with derivative bounds close to 1 and which depend on the ratio of the two marked length spectrum functions.Thus, we refine the results of Hamenstdt and Besson, Courtois and Gallot, which show M and N are isometric if their marked length spectra are equal.We also prove a similar result for closed negatively curved surfaces using the methods of Otal (1990) andPugh (1987).