Big islands in dispersing billiard-like potentials

We derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in smooth Hamiltonian approximations of scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, universality classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter ∈ → 0, islands of polynomial size in ∈ appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect.