Chaotic scattering by steep repelling potentials

Consider a classical two-dimensional scattering problem: a ray is scattered by a potential composed of several tall, repelling, steep mountains of arbitrary shape. We study when the traditional approximation of this nonlinear far-from-integrable problem by the corresponding simpler billiard problem, of scattering by hard-wall obstacles of similar shape, is justified. For one class of chaotic scatterers, named here regular Sinai scatterers, the scattering properties of the smooth system indeed limit to those of the billiards. For another class, the singular Sinai scatterers, these two scattering problems have essential differences: though the invariant set of such singular scatterers is hyperbolic (possibly with singularities), that of the smooth flow may have stable periodic orbits, even when the potential is arbitrarily steep. It follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards' deviation from a singular scatterer. Thus, even in this singular case, the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow. Finally, we see that corners have nontrivial and significant impact on the scattering functions.