Topology of Isoenergy Surfaces of Billiard Books Glued of Annuli

For an arbitrary billiard book glued from domains homeomorphic to annuli, it is shown that the isoenergy surface of the billiard dynamical system on such a table is homeomorphic to the direct product of the circle \(S^{1}\) and the sphere \(S^{2}\) with \(g\) handles. In the class of ordered billiard games introduced by Dragović et al. and modeled by them later by means of billiard books (algorithmically constructed from the billiard ordered game), a subclass of those games was found, the simulation of which is possible only by means of the billiard book subclass studied in this paper.