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 Tr. Mosk. Mat. Obs., 2015, Volume 76, Issue 2, Pages 287–308 (Mi mmo579)

Symmetric band complexes of thin type and chaotic sections which are not quite chaotic

I. Dynnikova, A. Skripchenkob

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

Abstract: In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic surface in the three-space with a plane direction such that the surface has a central symmetry, and the plane sections of the chosen direction are chaotic and consist of infinitely many connected components. Moreover, typical connected components of the sections have an asymptotic direction, which is due to the fact that the corresponding foliation on the surface in the 3-torus is not uniquely ergodic.
References: 25 entries.

Key words and phrases: band complex, Rips machine, Rauzy induction, measured foliation, ergodicity.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-12469 Dynasty Foundation

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English version:
Transactions of the Moscow Mathematical Society, 2015, 76:2, 251–269

UDC: 515.162
MSC: 57R30, 37E05, 37E25
Revised: 15.03.2015
Language:

Citation: I. Dynnikov, A. Skripchenko, “Symmetric band complexes of thin type and chaotic sections which are not quite chaotic”, Tr. Mosk. Mat. Obs., 76, no. 2, MCCME, M., 2015, 287–308; Trans. Moscow Math. Soc., 76:2 (2015), 251–269

Citation in format AMSBIB
\Bibitem{DynSkr15} \by I.~Dynnikov, A.~Skripchenko \paper Symmetric band complexes of thin type and chaotic sections which are not quite chaotic \serial Tr. Mosk. Mat. Obs. \yr 2015 \vol 76 \issue 2 \pages 287--308 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo579} \elib{https://elibrary.ru/item.asp?id=24850147} \transl \jour Trans. Moscow Math. Soc. \yr 2015 \vol 76 \issue 2 \pages 251--269 \crossref{https://doi.org/10.1090/mosc/246} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960081631} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. 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
2. W. P. Hooper, B. Weiss, “Rel leaves of the Arnoux–Yoccoz surfaces”, Sel. Math.-New Ser., 24:2 (2018), 875–934
3. 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
4. R. De Leo, A. Y. Maltsev, “Quasiperiodic dynamics and magnetoresistance in normal metals”, Acta Appl. Math., 162:1 (2019), 47–61
5. S. P. Novikov, R. De Leo, I. A. Dynnikov, A. Ya. Maltsev, “Theory of dynamical systems and transport phenomena in normal metals”, J. Exp. Theor. Phys., 129:4, SI (2019), 710–721
6. A. Ya. Maltsev, “The complexity classes of angular diagrams of the metal conductivity in strong magnetic fields”, J. Exp. Theor. Phys., 129:1 (2019), 116–138
7. R. De Leo, “A survey on quasiperiodic topology”, Advanced Mathematical Methods in Biosciences and Applications, Steam-H Science Technology Engineering Agriculture Mathematics & Health, eds. F. Berezovskaya, B. Toni, Springer, 2019, 53–88
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