Rotation Functions of Integrable Billiards As Orbital Invariants

Abstract— Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville’s theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko’s hypothesis. However, for some series of billiards, this hypothesis is true.