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

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

1999-09-28

arxiv.org/abs/math-ph/9909032v1

Russian Mathematical Surveys, vol. 54 (1999), no. 5, 1031--1032

Physics

Mathematical Physics

LATEX2e, 4 pages

Scientific paper

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.

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