The Levels of Quasiperiodic Functions on the plane, Hamiltonian Systems and Topology

Abstract

Topology of levels of the quasiperiodic functions with m=n+2 periods on the plane is studied. For the case of functions with m=4 periods full description is obtained for the open everywhere dense family of functions. This problem is equivalent to the study of Hamiltonian systems on the (n+2)-torus with constant rank 2 Poisson bracket. In the cases under investigation we proved that this system is topologically completely integrable in some natural sence where interesting integer-valued locally stable topological characteristics appear. The case of 3 periods has been extensively studied last years by the present author, Zorich, Dynnikov and Maltsev for the needs of solid state physics (''Galvanomagnetic Phenomena in Normal Metals''); The case of 4 periods might be useful for Quasicrystals.