Scattering rigidity with trapped geodesics

We prove that the flat product metric on ${D}^{n} \times {S}^{1} $ is scattering rigid where ${D}^{n} $ is the unit ball in ${ \mathbb{R} }^{n} $ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is the map $S: {U}^{+ } \partial M\rightarrow {U}^{- } \partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to ${ \gamma }_{V}^{\prime } ({T}_{0} )$ where ${\gamma }_{V} $ is the unit speed geodesic determined by $V$ and ${T}_{0} $ is the first positive value of $t$ (when it exists) such that ${\gamma }_{V} (t)$ again lies in the boundary. We show that any other Riemannian manifold $(M, \partial M, g)$ with boundary $\partial M$ isometric to $\partial ({D}^{n} \times {S}^{1} )$ and with the same scattering data must be isometric to ${D}^{n} \times {S}^{1} $. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M, \partial M, g)$ have measure zero in the unit tangent bundle.