Marked length spectrum rigidity for Anosov surfaces

Abstract

Let $\hat{I}\pounds$ be a smooth closed oriented surface of genus $\ge 2$. We prove that two metrics on $\hat{I}\pounds$ with the same marked length spectrum and Anosov geodesic flow are isometric via an isometry isotopic to the identity. The proof combines microlocal tools with the geometry of complex curves.

v2: We correct and simplify the proof of Lemma 3.11. v3: Added a corollary on the centralizer of Anosov geodesic flows. v4: Revision after referee process. To appear in Duke Mathematical Journal. 22 pages, 1 figure