Global stability of a class of discontinuous dual billiards

An infinite-parameter family of discontinuous area-preserving maps is studied, using geometrical methods. Necessary and sufficient conditions are determined for the existence of some bounding invariant sets, which guarantee global stability. It is shown that under some additional constraints, all orbits become periodic, most of them Lyapounov stable, and with a maximal period in any bounded domain of phase space. This yields a class of maps acting on a discrete phase space.