On Smooth Hamiltonian Flows Limiting to Ergodic Billiards

Suucient conditions are found so that a family of smooth Hamiltonian ows limits to a billiard ow as a parameter ! 0. This limit is proved to be C 1 near non-singular orbits and C 0 near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby Hamiltonian ows with elliptic islands. This implies that ergodicity may be lost for smooth potentials which are arbitrarily close to ergodic billiards. Thus, in some cases, anomoulous transport associated with stickiness to stability islands is expected