Lens rigidity with trapped geodesics in two dimensions

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat M\"obius strip are determined by their lens data. We also see by example that the flat M\"obius strip is not determined by it's scattering data. We then consider the case of negatively curved cylinders with convex boundary and show that they are lens rigid.