Integrable billiard systems realize toric foliations on lens spaces and the 3-torus

Abstract

An integrable billiard system on a book, a complex of several billiard sheets glued together along the common spine, is considered. Each sheet is a planar domain bounded by arcs of confocal quadrics; it is known that a billiard in such a domain is integrable. In a number of interesting special cases of such billiards the Fomenko-Zieschang invariants of Liouville equivalence (marked molecules ) turn out to describe nontrivial toric foliations on lens spaces and on the 3-torus, which are isoenergy manifolds for these billiards. Bibliography: 18 titles.