  In an ordinary billiard system trajectories of a Hamiltonian system are
elastically reflected after a collision with a hypersurface (scatterer). If the
scatterer is a submanifold of codimension more than one, we say that the
billiard is degenerate. Then collisions are rare. We study trajectories of
degenerate billiards which have an infinite number of collisions with the
scatterer. Degenerate billiards appear as limits of systems with elastic
reflections or as limits of systems with singularities in celestial mechanics.
We prove the existence of trajectories of such systems shadowing trajectories
of the corresponding degenerate billiards. The proofs are based on a version of
the method of an anti-integrable limit.
