I. A. Dynnikov, “Interval Identification Systems and Plane Sections of 3-Periodic Surfaces”, Geometry, topology, and mathematical physics. I, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 263, MAIK Nauka/Interperiodica, Moscow, 2008, 72–84; Proc. Steklov Inst. Math., 263 (2008), 65

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Volume 263, Pages 72–84 (Mi tm784)

This article is cited in 20 scientific papers (total in 20 papers)

Interval Identification Systems and Plane Sections of 3-Periodic Surfaces

I. A. Dynnikov

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

Full-text PDF (243 kB) Citations (20)

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Abstract: Interval identification systems is a notion that, on the one hand, generalizes interval exchange transformations and, on the other hand, describes special cases of such transformations. In the present paper we overview some elementary facts, address a few questions about interval identification systems, and describe explicitly systems that allow one to construct 3-periodic surfaces in the 3-space whose intersections with planes of a fixed direction have chaotic behavior. The problem of asymptotic behavior of plane sections of 3-periodic surfaces was posed by S. P. Novikov in 1982 and studied then by his students. One of the most interesting remaining open questions about such sections is reduced to the study of interval identification systems.

Received in April 2008

English version:
Proceedings of the Steklov Institute of Mathematics, 2008, Volume 263, Pages 65–77
DOI: https://doi.org/10.1134/S0081543808040068

Bibliographic databases:

UDC: 515.162

Language: Russian

Citation: I. A. Dynnikov, “Interval Identification Systems and Plane Sections of 3-Periodic Surfaces”, Geometry, topology, and mathematical physics. I, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 263, MAIK Nauka/Interperiodica, Moscow, 2008, 72–84; Proc. Steklov Inst. Math., 263 (2008), 65–77

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Linking options:

https://www.mathnet.ru/eng/tm784

https://www.mathnet.ru/eng/tm/v263/p72

This publication is cited in the following 20 articles:

A. Ya. Maltsev, S. P. Novikov, “Topology of dynamical systems on the Fermi surface and galvanomagnetic phenomena in normal metals”, Journal of Mathematical Physics, 65:7 (2024)
A. Ya. Maltsev, “On the Novikov Problem with a Large Number of Quasiperiods and Its Generalizations”, Proc. Steklov Inst. Math., 325 (2024), 163–176
A. S. Skripchenko, “Renormalization in one-dimensional dynamics”, Russian Math. Surveys, 78:6 (2023), 983–1021
A. Ya. Maltsev, “Lifshitz Transitions and Angular Conductivity Diagrams in Metals with Complex Fermi Surfaces”, J. Exp. Theor. Phys., 137:5 (2023), 706
Dynnikov I., Hubert P., Skripchenko A., “Dynamical Systems Around the Rauzy Gasket and Their Ergodic Properties”, Int. Math. Res. Notices, 2022
I. A. Dynnikov, A. Ya. Mal'tsev, S. P. Novikov, “Geometry of quasiperiodic functions on the plane”, Russian Math. Surveys, 77:6 (2022), 1061–1085
A. Ya. Maltsev, S. P. Novikov, “Open level lines of a superposition of periodic potentials on a plane”, Ann. Physics, 447 (2022), 169039–11
I. A. Dynnikov, A. Ya. Mal'tsev, S. P. Novikov, “Chaotic trajectories on fermi surfaces and nontrivial modes of behavior of magnetic conductivity”, J. Exp. Theor. Phys., 135 (2022), 240–254
Gutierrez-Romo R., Matheus C., “Lower Bounds on the Dimension of the Rauzy Gasket”, Bull. Soc. Math. Fr., 148:2 (2020), 321–327
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
De Leo R., “A Survey on Quasiperiodic Topology”, Advanced Mathematical Methods in Biosciences and Applications, Steam-H Science Technology Engineering Agriculture Mathematics & Health, eds. Berezovskaya F., Toni B., Springer International Publishing Ag, 2019, 53–88
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
Dynnikov I., Skripchenko A., “Minimality of interval exchange transformations with restrictions”, J. Mod. Dyn., 11 (2017), 219–248
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
Avila A., Hubert P., Skripchenko A., “Diffusion for chaotic plane sections of 3-periodic surfaces”, Invent. Math., 206:1 (2016), 109–146
Avila A., Hubert P., Skripchenko A., “On the Hausdorff dimension of the Rauzy gasket”, Bull. Soc. Math. Fr., 144:3 (2016), 539–568
McMullen C.T., “Cascades in the Dynamics of Measured Foliations”, Ann. Sci. Ec. Norm. Super., 48:1 (2015), 1–39
Trans. Moscow Math. Soc., 76:2 (2015), 251–269
Skripchenko A., “On Connectedness of Chaotic Sections of Some 3-Periodic Surfaces”, Ann. Glob. Anal. Geom., 43:3 (2013), 253–271
Skripchenko A., “Symmetric Interval Identification Systems of Order Three”, Discrete and Continuous Dynamical Systems, 32:2 (2012), 643–656

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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