Topological type of isoenergy surfaces of billiard books

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 $S^1\times S^2$.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.

integrable system, billiard book, Liouville equivalence, Fomenko-Zieschang invariant