Symmetric band complexes of thin type and chaotic sections which are not quite chaotic
HTML articles powered by AMS MathViewer
- by Ivan Dynnikov and Alexandra Skripchenko
- Trans. Moscow Math. Soc. 2015, 251-269
- DOI: https://doi.org/10.1090/mosc/246
- Published electronically: November 18, 2015
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
- Avila A., Hubert P., Skripchenko A., On the Hausdorff dimension of the Rauzy gasket, arXiv:1311.5361v2.
- Pierre Arnoux and Štěpán Starosta, The Rauzy gasket, Further developments in fractals and related fields, Trends Math., Birkhäuser/Springer, New York, 2013, pp. 1–23. MR 3184185, DOI 10.1007/978-0-8176-8400-6_{1}
- Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI 10.1007/BF01884300
- Thierry Coulbois, Fractal trees for irreducible automorphisms of free groups, J. Mod. Dyn. 4 (2010), no. 2, 359–391. MR 2672300, DOI 10.3934/jmd.2010.4.359
- H. S. M. Coxeter, Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues, Proc. London Math. Soc. (2) 43 (1937), no. 1, 33–62. MR 1575418, DOI 10.1112/plms/s2-43.1.33
- Roberto DeLeo and Ivan A. Dynnikov, Geometry of plane sections of the infinite regular skew polyhedron $\{4,6\mid 4\}$, Geom. Dedicata 138 (2009), 51–67. MR 2469987, DOI 10.1007/s10711-008-9298-1
- I. A. Dynnikov, A proof of the conjecture of S. P. Novikov on the semiclassical motion of an electron, Mat. Zametki 53 (1993), no. 5, 57–68 (Russian); English transl., Math. Notes 53 (1993), no. 5-6, 495–501. MR 1325615, DOI 10.1007/BF01208544
- I. A. Dynnikov, Semiclassical motion of the electron. A proof of the Novikov conjecture in general position and counterexamples, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 45–73. MR 1437157, DOI 10.1090/trans2/179/03
- I. A. Dynnikov, Interval covering systems and plane sections of 3-periodic surfaces, Tr. Mat. Inst. Steklova 263 (2008), no. Geometriya, Topologiya i Matematicheskaya Fizika. I, 72–84 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 263 (2008), no. 1, 65–77. MR 2599372, DOI 10.1134/S0081543808040068
- Ivan Dynnikov and Alexandra Skripchenko, On typical leaves of a measured foliated 2-complex of thin type, Topology, geometry, integrable systems, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 234, Amer. Math. Soc., Providence, RI, 2014, pp. 173–199. MR 3307148, DOI 10.1090/trans2/234/09
- Damien Gaboriau, Dynamique des systèmes d’isométries: sur les bouts des orbites, Invent. Math. 126 (1996), no. 2, 297–318 (French). MR 1411135, DOI 10.1007/s002220050101
- D. Gaboriau, G. Levitt, and F. Paulin, Pseudogroups of isometries of $\textbf {R}$ and Rips’ theorem on free actions on $\textbf {R}$-trees, Israel J. Math. 87 (1994), no. 1-3, 403–428. MR 1286836, DOI 10.1007/BF02773004
- Hideki Imanishi, On codimension one foliations defined by closed one-forms with singularities, J. Math. Kyoto Univ. 19 (1979), no. 2, 285–291. MR 545709, DOI 10.1215/kjm/1250522432
- Gilbert Levitt, La dynamique des pseudogroupes de rotations, Invent. Math. 113 (1993), no. 3, 633–670 (French, with English summary). MR 1231840, DOI 10.1007/BF01244321
- Maltsev A. Ya., Anomalous behavior of the electrical conductivity tensor in strong magnetic fields, JETP. vol. 85:5 (1997), 934–942.
- A. Ya. Maltsev and S. P. Novikov, Dynamical systems, topology, and conductivity in normal metals, J. Statist. Phys. 115 (2004), no. 1-2, 31–46. MR 2070088, DOI 10.1023/B:JOSS.0000019835.01125.92
- Curtis T. McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 151–183. MR 1953192, DOI 10.1007/s102400200001
- S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248 (Russian). MR 676612
- Alexandra Skripchenko, Symmetric interval identification systems of order three, Discrete Contin. Dyn. Syst. 32 (2012), no. 2, 643–656. MR 2837077, DOI 10.3934/dcds.2012.32.643
- Alexandra Skripchenko, On connectedness of chaotic sections of some 3-periodic surfaces, Ann. Global Anal. Geom. 43 (2013), no. 3, 253–271. MR 3027612, DOI 10.1007/s10455-012-9344-y
- William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
- Marcelo Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut. 19 (2006), no. 1, 7–100. MR 2219821, DOI 10.5209/rev_{R}EMA.2006.v19.n1.16621
- 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.
- Anton Zorich, Asymptotic flag of an orientable measured foliation, Séminaire de Théorie Spectrale et Géométrie, No. 11, Année 1992–1993, Sémin. Théor. Spectr. Géom., vol. 11, Univ. Grenoble I, Saint-Martin-d’Hères, 1993, pp. 113–131. MR 1715947, DOI 10.5802/tsg.135
- Anton Zorich, How do the leaves of a closed $1$-form wind around a surface?, Pseudoperiodic topology, Amer. Math. Soc. Transl. Ser. 2, vol. 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135–178. MR 1733872, DOI 10.1090/trans2/197/05
Bibliographic Information
- Ivan Dynnikov
- Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
- Email: dynnikov@mech.math.msu.su
- Alexandra Skripchenko
- Affiliation: Faculty of Mathematics, National Research University, Higher School of Economics, Moscow, Russia
- Email: sashaskrip@gm ail.com
- 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 - © Copyright 2015 I. Dynnikov, A. Skripchenko
- Journal: Trans. Moscow Math. Soc. 2015, 251-269
- MSC (2010): Primary 57R30, 37E05, 37E25
- DOI: https://doi.org/10.1090/mosc/246
- MathSciNet review: 3468067
Dedicated: On the occasion of Yu. Ilyashenko’s 70th birthday