Recurrent Motions in the Breathing Circle Billiard

The breathing circular billiard models the free motion of a point particle within a circular enclosure whose boundary undergoes periodic motion. It is assumed that the particle experiences elastic collisions with the boundary. Previous studies have demonstrated that when the boundary motion is sufficiently regular (specifically C7 smooth), the energy of the particle remains bounded due to the existence of invariant curves in the phase space. In this work, under the assumption that the motion of the boundary is C2, we prove that the set of initial conditions that give rise to escaping orbits has Lebesgue measure zero, imparting orbit stability to the breathing circular billiard.