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Transactions of the Moscow Mathematical Society

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

Authors: Ivan Dynnikov and Alexandra Skripchenko
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2015, 251-269
MSC (2010): Primary 57R30, 37E05, 37E25
Published electronically: November 18, 2015
MathSciNet review: 3468067
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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.

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  • 1. Avila A., Hubert P., Skripchenko A., On the Hausdorff dimension of the Rauzy gasket, arXiv:1311.5361v2.
  • 2. Arnoux P., Starosta Š., The Rauzy gasket, Further developments in fractals and related fields, Mathematical Foundations and Connections. New York: Birkhäuser/Springer, 2013. (Trends Math.; Vol. 13). P. 1-24. MR 3184185
  • 3. Bestvina M., Feighn M., Stable actions of groups on real trees, Invent. Math. 1995. Vol. 121, No. 2. P. 287-321. MR 1346208 (96h:20056)
  • 4. Coulbois T., Fractal trees for irreducible automorphisms of free groups, J. Mod. Dyn. 2010. Vol. 4, No. 2. P. 359-391. MR 2672300 (2011i:20037)
  • 5. Coxeter H. S. M., Regular skew polyhedra in three and four dimensions and their topological analogues, Proc. London Math. Soc. 1937. Vol. 43, No. 1. P. 33-62. MR 1575418
  • 6. De Leo R., Dynnikov I., Geometry of plane sections of the infinite regular skew polyhedron $ \{4,6\mid 4\},$ Geom. Dedic. 2009. Vol. 138, No. 1. P. 51-67. MR 2469987 (2010a:37085)
  • 7. Dynnikov I., A proof of the conjecture of S. P. Novikov on semiclassical motion of an electron, Math. Notes. 1993. Vol. 53, No. 6. P. 495-501. MR 1325615 (96a:58066)
  • 8. Dynnikov I. A., Semiclassical motion of the electron. A proof of the Novikov conjecture in general position and counterexamples, Solitons, geometry, and topology: on the crossroad. Providence, RI: AMS, 1997. (AMS Transl. Ser. 2; Vol. 179). P. 45-73. MR 1437157 (99k:57074)
  • 9. Dynnikov I., Interval identification systems and plane sections of 3-periodic surfaces, Proceedings of the Steklov Institute of Mathematics. 2008. Vol. 263. P. 65-77. MR 2599372 (2011g:37004)
  • 10. Dynnikov I., Skripchenko A., On typical leaves of a measured foliated 2-complex of thin type, Topology, geometry, integrable systems, and mathematical physics. Providence, RI: AMS, 2014. (AMS Transl. Ser. 2; Vol. 234). P. 173-199. MR 3307148
  • 11. Gaboriau D., Dynamique des systèmes d'isométries: sur les bouts des orbits, Invent. Math. 1996. Vol. 126, No. 2. P. 297-318. MR 1411135 (97i:57028)
  • 12. Gaboriau D., Levitt G., Paulin F., Pseudogroups of isometries of $ \mathbb{R}$ and Rips' theorem on free actions on $ \mathbb{R}$-trees, Israel J. Math. 1994. Vol. 87. P. 403-428. MR 1286836 (95e:20042)
  • 13. Imanishi H., On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ. 1979. Vol. 19. P. 285-291. MR 545709 (80k:57050)
  • 14. Levitt G., La dynamique des pseudogroupes de rotations, Invent. Math. vol. 113 (1993), 633-670. MR 1231840 (94k:58158)
  • 15. Maltsev A. Ya., Anomalous behavior of the electrical conductivity tensor in strong magnetic fields, JETP. vol. 85:5 (1997), 934-942.
  • 16. Maltsev A. Ya., Novikov S. P., Dynamical systems, topology, and conductivity in normal metals, J. Stat. Phys. vol. 115 (2004), 31-46. MR 2070088 (2006b:82128)
  • 17. McMullen C. T., Coxeter groups, Salem numbers and the Hilbert metric, Publications Mathématiques de l'IHÉS. vol. 95 (2002), 151-183. MR 1953192 (2004b:20054)
  • 18. Novikov S. P., The Hamiltonian formalism and multivalued analogue of Morse theory (in Russian), Uspekhi Mat. Nauk. vol. 37:5 (1982), 3-49. Translated in Russ. Math. Surv. vol. 37:5 (1982), 1-56. MR 676612 (84h:58032)
  • 19. Skripchenko A., Symmetric interval identification systems of order 3, Disc. Cont. Dyn. Syst. vol. 32:2 (2012), 643-656. MR 2837077
  • 20. Skripchenko A., On connectedness of chaotic sections of some 3-periodic surfaces, Ann. Glob. Anal. Geom. vol. 43 (2013), 253-271. MR 3027612
  • 21. Veech W. A., Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2). vol. 115:1 (1982), 201-242. MR 644019 (83g:28036b)
  • 22. Viana M., Ergodic theory of interval exchange maps, Rev. Mat. Complut. vol. 19:1 (2006), 7-100. MR 2219821 (2007f:37002)
  • 23. Zorich A., A Problem of Novikov on the semiclassical motion of an electron in a uniform almost rational magnetic field, Russ. Math. Surv. vol. 39:5 (1984), 287-288.
  • 24. Zorich A., Asymptotic flag of an orientable measured foliation, Séminaire de théorie spectrale et géométrie. vol. 11 (1992-1993), 113-131. MR 1715947
  • 25. Zorich A., How do the leaves of a closed 1-form wind around a surface?, Pseudoperiodic topology. Providence, RI: AMS, 1999. (AMS Transl. Ser. 2; Vol. 197). P. 135-178. MR 1733872 (2001c:57019)

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Additional Information

Ivan Dynnikov
Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Alexandra Skripchenko
Affiliation: Faculty of Mathematics, National Research University, Higher School of Economics, Moscow, Russia
Email: sashaskrip@gm

Keywords: Band complex, Rips machine, Rauzy induction, measured foliation, ergodicity
Published electronically: November 18, 2015
Additional Notes: The first author was supported in part by the Russian Foundation for Basic Research (grant No.13-01-12469)
The second author is partially supported by Lavrentiev Prix and by the Dynasty Foundation
The authors thank their anonymous referee for a careful reading of their paper and for a number of helpful remarks
Dedicated: On the occasion of Yu.Ilyashenko’s 70th birthday
Article copyright: © Copyright 2015 I. Dynnikov, A. Skripchenko

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