Bifurcations of Billiard Books Modelling Seifert Fibrations with an Arbitrary Set of Special Fibers

Abstract

This paper is devoted to developing methods for topological modelling of integrable Hamiltonian systems using integrable billiards. Progress is made in proving Fomenko’s billiard conjecture. Specifically, integrable generalized billiards realize three-dimensional Seifert manifolds with an arbitrary set of singular fibers of types \((p_i, q_i)\). In other words, the types of singular fibers do not in themselves obstruct the realization of a foliation by billiards. The two-dimensional base of the corresponding Seifert fibration is glued together by multisaddle (non-Bott) 2-atoms. As a consequence, a proof of the “maximal” version of Section \(A\) of Fomenko’s conjecture is obtained: billiard books realize arbitrary generalizations of Bott 3-atoms with a circle action and a base consisting of non-Morse multisaddle 2-atoms.