# Fomenko-Zieshang invariants help to model complex integrable Hamiltonian systems by integrable billiards.

The first part of the talk will be devoted to bases, main ideas and constructions of Fomenko-Zieschang theory of invariants of integrable Hamiltonian systems. In fact, such graph invariants help effectively describe the topology of phase space foliated on the closures of solutions of integrable systems. This invariant ia complete, so two such systems have fiber-wise diffeomorphic foliations if and only if their invariants coincide. Some their extension also classifies structures of trajectories of such systems in the sense of topological (orbital) equivalence.

In the second part the piece-wise smooth case will be considered. Class of integrable billiards in flat domains was generalized in recent time by gluing elementary domains by some of their common boundaries.

Calculation of their Fomenko-Zieschang invariants shows that smooth integrable systems in mechanics and geodesic flows on orientable 2-surfaces often have the same structure of closures of trajectories.

Even if billiard domain is a some complex, it is more simple to see interesting effects (bifurcation of Liouville tori, periodic critical trajectories, their stability) than in integrable systems of higher degree.

So, it allows to say that "visual" billiard system in a suitable domains gives us a chance to describe the behavior of a complicated integrable system.