  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.
