Fomenko conjecture: realization of Liouville foliations by integrable billiards

Establish that the topology of Liouville foliations of smooth and real-analytic integrable Hamiltonian systems can be realized by integrable billiard systems, thereby confirming the universality of billiard dynamics in reproducing Liouville foliation topologies.

Background

The paper reviews the topological classification of integrable systems using Fomenko graphs and Fomenko–Zieschang invariants and highlights the central role of billiards within this framework. The authors reference the Fomenko conjecture as a guiding open problem asserting a universality property: that integrable billiards can realize the topology of Liouville foliations of integrable Hamiltonian systems.

Within the context of this work, the authors introduce magic billiards and compute Fomenko graphs for several integrable cases with elliptical boundaries, illustrating how billiard models can reproduce diverse Liouville foliation topologies, thereby motivating and connecting to the conjecture.