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 Uspekhi Mat. Nauk, 2005, Volume 60, Issue 1(361), Pages 3–28 (Mi umn1386)

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

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].

DOI: https://doi.org/10.4213/rm1386

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English version:
Russian Mathematical Surveys, 2005, 60:1, 1–26

Bibliographic databases:

Document Type: Article
UDC: 515.16
MSC: Primary 37N20, 37J05; Secondary 37E35, 37C55, 70K43, 82D35, 82D25

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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• http://mi.mathnet.ru/eng/umn1386
• https://doi.org/10.4213/rm1386
• http://mi.mathnet.ru/eng/umn/v60/i1/p3

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This publication is cited in the following articles:
1. S. P. Novikov, “Topology of generic Hamiltonian foliations on Riemann surfaces”, Mosc. Math. J., 5:3 (2005), 633–667
2. Grinevich P.G., Santini P.M., “Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $\mu^2=\nu^n-1$, $n\in\mathbb Z$: ergodicity, isochrony and fractals”, Phys. D, 232:1 (2007), 22–32
3. V. V. Kozlov, “Dynamical Systems with Multivalued Integrals on a Torus”, Proc. Steklov Inst. Math., 256 (2007), 188–205
4. Birindelli I., Valdinoci E., “The Ginzburg-Landau equation in the Heisenberg group”, Commun. Contemp. Math., 10:5 (2008), 671–719
5. Novikov S.P., “Dynamical Systems and Differential Forms. Low Dimensional Hamiltonian Systems”, Geometric and Probabilistic Structures in Dynamics, Contemporary Mathematics Series, 469, 2008, 271–287
6. A. B. Antonevich, A. N. Buzulutskaya (Glaz), “Almost-Periodic Algebras and Their Automorphisms”, Math. Notes, 102:5 (2017), 610–622
7. A. Ya. Maltsev, S. P. Novikov, “The theory of closed 1-forms, levels of quasiperiodic functions and transport phenomena in electron systems”, Proc. Steklov Inst. Math., 302 (2018), 279–297
8. A. Ya. Maltsev, S. P. Novikov, “Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter”, Russian Math. Surveys, 74:1 (2019), 141–173
9. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140
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