Billiard with slipping by an arbitrary rational angle

Abstract

The class of billiards in a disc with slipping along the boundary circle by an angle commensurable with $\pi$ is considered. For such billiards it is shown that an isoenergy surface of the system is homeomorphic to a lens space $L(q,p)$ with parameters satisfying $0 < p <q$. The set of pairs $(q, p)$ such that there exists a billiard in a disc realizing the corresponding lens space $L(q,p)$ is described in terms of solutions of a linear Diophantine equation in two variables. This result also holds for planar billiards with slipping in simply connected domains with smooth boundary, that is, it is not confined to the integrable case. Bibliography: 30 titles.