Given a closed negatively curved manifold, we consider the extent to which dynamical data associated to its closed geodesics (equivalently, periodic orbits of its geodesic flow) determines the underlying metric up to isometry. For instance, the lengths of closed geodesics, marked by their free homotopy classes, are conjectured to characterize the underlying metric up to isometry. In this talk, we consider a dynamically flavored variant of this marked length spectrum rigidity problem. We introduce the marked Poincare determinant, which associates to each free homotopy class of closed curves a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. Our main result is that near hyperbolic metrics in dimension 3, this invariant determines the metric up to homothety. This is joint work with Erchenko, Humbert, Lefeuvre, and Wilkinson.
