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

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

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
\Bibitem{Dyn08} \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 \mathnet{http://mi.mathnet.ru/tm784} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2599372} \zmath{https://zbmath.org/?q=an:1231.37024} \elib{https://elibrary.ru/item.asp?id=11640635} \transl \jour Proc. Steklov Inst. Math. \yr 2008 \vol 263 \pages 65--77 \crossref{https://doi.org/10.1134/S0081543808040068} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000263177700005} \elib{https://elibrary.ru/item.asp?id=13592058} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-59849105280} 

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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. Skripchenko A., “Symmetric Interval Identification Systems of Order Three”, Discrete and Continuous Dynamical Systems, 32:2 (2012), 643–656
2. Skripchenko A., “On Connectedness of Chaotic Sections of Some 3-Periodic Surfaces”, Ann. Glob. Anal. Geom., 43:3 (2013), 253–271
3. McMullen C.T., “Cascades in the Dynamics of Measured Foliations”, Ann. Sci. Ec. Norm. Super., 48:1 (2015), 1–39
4. Trans. Moscow Math. Soc., 76:2 (2015), 251–269
5. Avila A., Hubert P., Skripchenko A., “Diffusion for chaotic plane sections of 3-periodic surfaces”, Invent. Math., 206:1 (2016), 109–146
6. Avila A., Hubert P., Skripchenko A., “On the Hausdorff dimension of the Rauzy gasket”, Bull. Soc. Math. Fr., 144:3 (2016), 539–568
7. Dynnikov I., Skripchenko A., “Minimality of interval exchange transformations with restrictions”, J. Mod. Dyn., 11 (2017), 219–248
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
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
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
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