Elliptic Islands Appearing in Near-ergodic Ows

It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in thènearby' Hamiltonian ows; i.e., in a family of two-degrees-of-freedom smooth Hamiltonian ows which converge to the singular billiard ow smoothly where the billiard ow is smooth and continuously where it is continuous. Such Hamiltonians exist; Indeed , suucient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard ows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.