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 Uspekhi Mat. Nauk, 1999, Volume 54, Issue 1(325), Pages 21–60 (Mi umn116)

The geometry of stability regions in Novikov's problem on the semiclassical motion of an electron

I. A. Dynnikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

DOI: https://doi.org/10.4213/rm116

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English version:
Russian Mathematical Surveys, 1999, 54:1, 21–59

Bibliographic databases:

UDC: 517.98
MSC: Primary 57R70; Secondary 58E05, 57R30, 81Q20, 81V10, 78A35

Citation: I. A. Dynnikov, “The geometry of stability regions in Novikov's problem on the semiclassical motion of an electron”, Uspekhi Mat. Nauk, 54:1(325) (1999), 21–60; Russian Math. Surveys, 54:1 (1999), 21–59

Citation in format AMSBIB
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• https://doi.org/10.4213/rm116
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. P. Novikov, “Levels of quasiperiodic functions on a plane, and Hamiltonian systems”, Russian Math. Surveys, 54:5 (1999), 1031–1032
2. R. De Leo, “The existence and measure of ergodic foliations in Novikov's problem of the semiclassical motion of an electron”, Russian Math. Surveys, 55:1 (2000), 166–168
3. Novikov S.P., “1. Classical and modern topology 2. Topological phenomena in real world physics”, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal., 2000, 406–424
4. R. De Leo, “Characterization of the set of “ergodic directions” in Novikov's problem of quasi-electron orbits in normal metals”, Russian Math. Surveys, 58:5 (2003), 1042–1043
5. De Leo R., “Numerical analysis of the Novikov problem of a normal metal in a strong magnetic field”, SIAM J. Appl. Dyn. Syst., 2:4 (2003), 517–545
6. Maltsev A.Yu., Novikov S.P., “Quasiperiodic functions and dynamical systems in quantum solid state physics”, Bull. Braz. Math. Soc. (N.S.), 34:1 (2003), 171–210
7. Maltsev A.Ya., “Quasiperiodic functions theory and the superlattice potentials for a two-dimensional electron gas”, J. Math. Phys., 45:3 (2004), 1128–1149
8. De Leo R., “Topological effects in the magnetoresi stance of Au and Ag”, Phys. Lett. A, 332:5-6 (2004), 469–474
9. Maltsev A.Yu., Novikov S.P., “Dynamical systems, topology, and conductivity in normal metals”, J. Statist. Phys., 115:1-2 (2004), 31–46
10. I. A. Dynnikov, S. P. Novikov, “Topology of quasi-periodic functions on the plane”, Russian Math. Surveys, 60:1 (2005), 1–26
11. R. De Leo, “Proof of Dynnikov's conjecture on the location of stability zones in the Novikov problem on planar sections of periodic surfaces”, Russian Math. Surveys, 60:3 (2005), 566–567
12. De Leo R., “First-principles generation of stereographic maps for high-field magneto resistance in normal metals: An application to Au and Ag”, Physica B: Physics of Condensed Matter, 362:1-4 (2005), 62–75
13. Gelbukh I., “Presence of minimal components in a Morse form foliation”, Differential Geom. Appl., 22:2 (2005), 189–198
14. De Leo R., “Topology of plane sections of periodic polyhedra with an application to the truncated octahedron”, Experiment. Math., 15:1 (2006), 109–124
15. Maltsev A.Y., Novikov S.P., “Topology, quasiperiodic functions, and the transport phenomena”, Topology in Condensed Matter, Springer Series in Solid-State Sciences, 150, 2006, 31–59
16. Zorich A., “Flat surfaces”, Frontiers in Number Theory, Physics and Geometry I - ON RANDOM MATRICES, ZETA FUNCTIONS, AND DYNAMICAL SYSTEMS, 2006, 439–585
17. R. De Leo, I. A. Dynnikov, “An example of a fractal set of plane directions having chaotic intersections with a fixed 3-periodic surface”, Russian Math. Surveys, 62:5 (2007), 990–992
18. Millionschikov D.V., “Multi-valued functionals, one-forms and deformed de Rham complex”, Topology in Molecular Biology, Biological and Medical Physics, Biomedical Engineering, 2007, 189–208
19. I. A. Dynnikov, “Interval Identification Systems and Plane Sections of 3-Periodic Surfaces”, Proc. Steklov Inst. Math., 263 (2008), 65–77
20. DeLeo R., Dynnikov I.A., “Geometry of plane sections of the infinite regular skew polyhedron ${4,6\mid 4}$”, Geom. Dedicata, 138:1 (2009), 51–67
21. Maltsev A.Ya., “Oscillation Phenomena and Experimental Determination of Exact Mathematical Stability Zones For Magneto-Conductivity in Metals Having Complicated Fermi Surfaces”, J. Exp. Theor. Phys., 125:5 (2017), 896–905
22. Maltsev A.Ya., “On the Analytical Properties of the Magneto-Conductivity in the Case of Presence of Stable Open Electron Trajectories on a Complex Fermi Surface”, J. Exp. Theor. Phys., 124:5 (2017), 805–831
23. 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
24. Maltsev A.Ya., “The Second Boundaries of Stability Zones and the Angular Diagrams of Conductivity For Metals Having Complicated Fermi Surfaces”, J. Exp. Theor. Phys., 127:6 (2018), 1087–1111
25. 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
26. De Leo R. Maltsev A.Y., “Quasiperiodic Dynamics and Magnetoresistance in Normal Metals”, Acta Appl. Math., 162:1 (2019), 47–61
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