Topological type of isoenergy surfaces of billiard books

Abstract

The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products . The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms — that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems — are calculated for several integrable billiards of this type. Bibliography: 14 titles.