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

TL;DR

This paper investigates the topology of quasiperiodic functions with multiple periods on the plane, linking it to Hamiltonian systems on tori, and provides a complete description for functions with four periods, revealing topological integrability and stable characteristics.

Contribution

It offers a full topological classification of quasiperiodic functions with four periods and connects these results to Hamiltonian systems with applications in physics.

Findings

Complete description for quasiperiodic functions with four periods.

Identification of topological integrability in certain Hamiltonian systems.

Discovery of stable topological characteristics in these systems.

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.