Causal holography and applications to inverse scattering problems

For a given smooth compact manifold $M$, we introduce a massive class $\mathcal G(M)$ of Riemannian metrics. We call them metrics of the gradient type. For such metrics $g$, the geodesic flow $v(g)$ on the spherical tangent bundle $SM \to M$ admits a Lyapunov function. Moreover, for every $g \in \mathcal G(M)$, the geodesic scattering along the boundary $\d M$ can be expressed in terms of a discontinuous map
$C_{v(g)}: \d_1^+(SM) \to \d_1^-(SM)$. It acts from a domain $\d_1^+(SM)$ in the boundary $\d(SM)$ to the complementary domain $\d_1^-(SM)$, both domains being diffeomorphic.

We prove that, for a boundary generic metric $g$ of the gradient type, the scattering map $C_{v(g)}$ allows for a reconstruction of $SM$ and of the geodesic flow on it, up to a homeomorphism (often a diffeomorphism). Also, for such $g$, the knowledge of the scattering map $C_{v(g)}$ allows to reconstruct the homology of $M$, the Gromov simplicial semi-norm on it, and the fundamental group of $M$.

We aim to understand the constraints on $(M, g)$ under which the scattering map allows for a reconstruction of $M$ and of the metric $g$ on it. In particular, we consider a closed Riemannian $n$-manifold $(N, g)$ which is locally symmetric and of a negative sectional curvature. Let $M$ is obtained from $N$ by removing several $n$-balls so that the metric $g|_M$ is boundary generic and of the gradient type. Then we prove that the scattering map $C_{v(g|_M)}$ makes it possible to recover $N$ and the metric $g$ on it.