The boundary rigidity problem asks if the boundary distance function on a simple Riemannian manifold determines the metric. For non simple manifolds with boundary or even for closed Riemannian manifolds, there are corresponding problems named lens rigidity and marked length spectrum rigidity. The general question is essentially reduced to knowing if a conjugacy between two geodesic flows on the unit tangent bundles necessarily comes from an isometry on the underlying Riemannian metrics.

The introduction of microlocal methods to understand the regularity of solutions of transport equations, of invariant distributions for the geodesic flow, has been key in the resolution of such problems that naturally arise when the geodesic flow is chaotic (hyperbolic). We will review recent results in this direction, in collaborations with Bonthonneau, Cekic, Jezequel, Lefeuvre and Paternain.