Billiard Sequences and the Property of Splittability of Integrable Hamilton Systems

Abstract

The paper establishes the property of splittability of billiard boundary sequences in n dimensional cube into subsequences of fractional parts. This reveals a new property of integrable and weak perturbated Hamilton systems: under a simple assumption, the boundary motion of elliptic orbits on stable KAM tori, if considering in cartesian coordinates, can be splitted into a countable set of discrete rotations. The rate of the split process, expressed in terms of some exceptional sets density, in dependence of number-theoretical characteristics of the orbits frequencies, is also examined.