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Russian Mathematical Surveys, 2005, Volume 60, Issue 1, Pages 1–26
DOI: https://doi.org/10.1070/RM2005v060n01ABEH000806
(Mi rm1386)
 

This article is cited in 22 scientific papers (total in 22 papers)

Topology of quasi-periodic functions on the plane

I. A. Dynnikova, S. P. Novikovbc

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
c University of Maryland
References:
Abstract: In this paper the topological theory of quasi-periodic functions on the plane is presented. The development of this theory was started (in another terminology) by the Moscow topology group in the early 1980s, motivated by needs of solid state physics which led to the necessity of investigating a special (non-generic) case of Hamiltonian foliations on Fermi surfaces with a multivalued Hamiltonian function [1]. These foliations turned out to have unexpected topological properties, discovered in the 1980s ([2], [3]) and 1990s ([4]–[6]), which led finally to non-trivial physical conclusions ([7], [8]) by considering the so-called geometric strong magnetic field limit [9]. A reformulation of the problem in terms of quasi-periodic functions and an extension to higher dimensions in 1999 [10] produced a new and fruitful approach. One can say that for monocrystalline normal metals in a magnetic field the semiclassical trajectories of electrons in the quasi-momentum space are exactly the level curves of a quasi-periodic function with three quasi-periods which is the restriction of the dispersion relation to the plane orthogonal to the magnetic field. The general study of topological properties of level curves for quasi-periodic functions on the plane with arbitrarily many quasi-periods began in 1999 when some new ideas were formulated in the case of four quasi-periods [10]. The last section of this paper contains a complete proof of these results based on the technique developed in [11] and [12]. Some new physical applications of the general problem were found recently [13].
Received: 26.12.2004
Russian version:
Uspekhi Matematicheskikh Nauk, 2005, Volume 60, Issue 1(361), Pages 3–28
DOI: https://doi.org/10.4213/rm1386
Bibliographic databases:
Document Type: Article
UDC: 515.16
MSC: Primary 37N20, 37J05; Secondary 37E35, 37C55, 70K43, 82D35, 82D25
Language: English
Original paper language: Russian
Citation: I. A. Dynnikov, S. P. Novikov, “Topology of quasi-periodic functions on the plane”, Uspekhi Mat. Nauk, 60:1(361) (2005), 3–28; Russian Math. Surveys, 60:1 (2005), 1–26
Citation in format AMSBIB
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\paper Topology of quasi-periodic functions on the plane
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 1(361)
\pages 3--28
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\jour Russian Math. Surveys
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\vol 60
\issue 1
\pages 1--26
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  • https://www.mathnet.ru/eng/rm/v60/i1/p3
  • This publication is cited in the following 22 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    Russian version PDF:603
    English version PDF:37
    References:112
    First page:6
     
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