Topology of quasiperiodic functions on the plane

Abstract

The article describes a topological theory of quasiperiodic functions on the plane. The development of this theory was started (in different terminology) by the Moscow topology group in early 1980s. It was motivated by the needs of solid state physics, as a partial (nongeneric) case of Hamiltonian foliations of Fermi surfaces with multivalued Hamiltonian function. The unexpected discoveries of their topological properties that were made in 1980s and 1990s have finally led to nontrivial physical conclusions along the lines of the so-called geometric strong magnetic field limit. A very fruitful new point of view comes from the reformulation of that problem in terms of quasiperiodic functions and an extension to higher dimensions made in 1999. One may say that, for single crystal normal metals put in a magnetic field, the semiclassical trajectories of electrons in the space of quasimomenta are exactly the level lines of the quasiperiodic function with three quasiperiods that is the dispersion relation restricted to a plane orthogonal to the magnetic field. General studies of the topological properties of levels of quasiperiodic functions on the plane with any number of quasiperiods were started in 1999 when certain ideas were formulated for the case of four quasiperiods. The last section of this work contains a complete proof of these results. Some new physical applications of the general problem were found recently.

latex2e, 27 pages, 7 figures

References in corpus (4)

• Dynamical Systems, Topology and Conductivity in Normal Metals in strong magnetic fields
• Dynamical Systems, Topology and Conductivity in Normal Metals
• Quasiperiodic functions theory and the superlattice potentials for a two-dimensional electron gas
• Topology, Quasiperiodic functions and the Transport phenomena

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• Features of the motion of ultracold atoms in quasiperiodic potentials
• Topology of the Generic Hamiltonian Foliations on the Riemann Surface
• On the Novikov problem with a large number of quasiperiods and its generalizations
• The theory of closed 1-forms, levels of quasiperiodic functions and transport phenomena in electron systems
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• The Ginzburg-Landau equation in the Heisenberg group
• Topology of Foliations given by the real part of holomorphic 1-forms