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Tr. Mat. Inst. Steklova, 2008, Volume 263, Pages 72–84 (Mi tm784)  

This article is cited in 10 scientific papers (total in 10 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

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2008, 263, 65–77

Bibliographic databases:

UDC: 515.162
Received in April 2008

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, Tr. Mat. Inst. Steklova, 263, MAIK Nauka/Interperiodica, Moscow, 2008, 72–84; Proc. Steklov Inst. Math., 263 (2008), 65–77

Citation in format AMSBIB
\by I.~A.~Dynnikov
\paper Interval Identification Systems and Plane Sections of 3-Periodic Surfaces
\inbook Geometry, topology, and mathematical physics.~I
\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2008
\vol 263
\pages 72--84
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2008
\vol 263
\pages 65--77

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    This publication is cited in the following articles:
    1. Skripchenko A., “Symmetric Interval Identification Systems of Order Three”, Discrete and Continuous Dynamical Systems, 32:2 (2012), 643–656  crossref  mathscinet  zmath  isi  scopus
    2. Skripchenko A., “On Connectedness of Chaotic Sections of Some 3-Periodic Surfaces”, Ann. Glob. Anal. Geom., 43:3 (2013), 253–271  crossref  mathscinet  zmath  isi  elib  scopus
    3. McMullen C.T., “Cascades in the Dynamics of Measured Foliations”, Ann. Sci. Ec. Norm. Super., 48:1 (2015), 1–39  crossref  mathscinet  zmath  isi  elib  scopus
    4. Trans. Moscow Math. Soc., 76:2 (2015), 251–269  mathnet  crossref  elib
    5. Avila A., Hubert P., Skripchenko A., “Diffusion for chaotic plane sections of 3-periodic surfaces”, Invent. Math., 206:1 (2016), 109–146  crossref  mathscinet  zmath  isi  scopus
    6. Avila A., Hubert P., Skripchenko A., “On the Hausdorff dimension of the Rauzy gasket”, Bull. Soc. Math. Fr., 144:3 (2016), 539–568  crossref  mathscinet  zmath  isi  scopus
    7. Dynnikov I., Skripchenko A., “Minimality of interval exchange transformations with restrictions”, J. Mod. Dyn., 11 (2017), 219–248  crossref  mathscinet  isi  scopus
    8. 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  crossref  isi  scopus
    9. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    10. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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