  We develop a framework for dealing with smooth approximations to billiards
with corners in the two-dimensional setting. Let a polygonal trajectory in a
billiard start and end up at the same billiard's corner point. We prove that
smooth Hamiltonian flows which limit to this billiard have a nearby periodic
orbit if and only if the polygon angles at the corner are ''acceptable''. The
criterion for a corner polygon to be acceptable depends on the smooth potential
behavior at the corners, which is expressed in terms of a {scattering
function}. We define such an asymptotic scattering function and prove the
existence of it, explain how it can be calculated and predict some of its
properties. In particular, we show that it is non-monotone for some potentials
in some phase space regions. We prove that when the smooth system has a
limiting periodic orbit it is hyperbolic provided the scattering function is
not extremal there. We then prove that if the scattering function is extremal,
the smooth system has elliptic periodic orbits limiting to the corner polygon,
and, furthermore, that the return map near these periodic orbits is conjugate
to a small perturbation of the Henon map and therefore has elliptic islands. We
find from the scaling that the island size is typically algebraic in the
smoothing parameter and exponentially small in the number of reflections of the
polygon orbit.
