Ordered billiard games and topological properties of billiard books

We discuss the connection between the construction of an ordered billiard game introduced earlier by Dragovic and Radnovic and the class of billiard books proposed by Vedyushkina. In this paper, we propose a generalization of the concept of realization of a certain game using a billiard book and prove an analogue of the Dragovic–Radnovic theorem for such a realization. We present recent results by the authors, Tyurina, and Zav'ialov on topological properties of isoenergy manifolds of circular billiard books and topological invariants of specific series of elliptic billiard books.