Orbital invariants of billiards and linearly integrable geodesic flows

Orbital invariants of integrable topological billiards with two degrees of freedom are discovered and calculated in the case of constant energy of the system. These invariants (rotation vectors) are calculated in terms of rotation functions on one-parameter families of Liouville 2-tori. An analogue of Liouville's theorem is proved for a piecewise smooth billiard in a neighbourhood of a regular level. Action-angle variables are introduced. A general formula for rotation functions is obtained. There was a conjecture due to Fomenko that the rotation functions of topological billiards are monotone. This conjecture was confirmed for many important systems, but interesting billiard systems with nonmonotone rotation functions were also discovered. In particular, orbital invariants of billiard books realizing (up to Liouville equivalence) linearly integrable geodesic flows of 2-dimensional surfaces are calculated. After a suitable change of the parameters of the flows these functions become monotone.