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.
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
\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}
Linking options:
https://www.mathnet.ru/eng/mmo579
https://www.mathnet.ru/eng/mmo/v76/i2/p287
This publication is cited in the following 19 articles:
A. Ya. Maltsev, “On the Novikov Problem with a Large Number of Quasiperiods and Its Generalizations”, Proc. Steklov Inst. Math., 325 (2024), 163–176
A. Ya. Mal'tsev, “OSOBENNOSTI τ -PRIBLIZhENIYa DLYa KhAOTIChESKIKh ELEKTRONNYKh TRAEKTORIY NA SLOZhNYKh POVERKhNOSTYaKh FERMI”, Žurnal èksperimentalʹnoj i teoretičeskoj fiziki, 166:3 (2024)
A. S. Skripchenko, “Renormalization in one-dimensional dynamics”, Russian Math. Surveys, 78:6 (2023), 983–1021
A. Ya. Maltsev, “Lifshitz Transitions and Angular Conductivity Diagrams in Metals with Complex Fermi Surfaces”, J. Exp. Theor. Phys., 137:5 (2023), 706
Ivan Dynnikov, Pascal Hubert, Alexandra Skripchenko, “Dynamical Systems Around the Rauzy Gasket and Their Ergodic Properties”, Int. Math. Res. Not. IMRN, 2023:8 (2023), 6461–6503
A. Ya Mal'tsev, “Perekhody Lifshitsa i uglovye diagrammy provodimosti v metallakh so slozhnymi poverkhnostyami Fermi”, Zhurnal eksperimentalnoi i teoreticheskoi fiziki, 164:5 (2023), 817
I. A. Dynnikov, A. Ya. Mal'tsev, S. P. Novikov, “Geometry of quasiperiodic functions on the plane”, Russian Math. Surveys, 77:6 (2022), 1061–1085
I. A. Dynnikov, A. Ya. Mal'tsev, S. P. Novikov, “Chaotic trajectories on fermi surfaces and nontrivial modes of behavior of magnetic conductivity”, J. Exp. Theor. Phys., 135 (2022), 240–254
Dynnikov I., Maltsev A., “Features of the Motion of Ultracold Atoms in Quasiperiodic Potentials”, J. Exp. Theor. Phys., 133:6 (2021), 711–736
O. Paris-Romaskevich, “Tiling billiards and Dynnikov's helicoid”, Trans. Moscow Math. Soc., 82 (2021), 133–147
A. Ya. Maltsev, “Reconstructions of the electron dynamics in magnetic field and the geometry of complex fermi surfaces”, J. Exp. Theor. Phys., 131:6 (2020), 988–1020
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
R. De Leo, A. Y. Maltsev, “Quasiperiodic dynamics and magnetoresistance in normal metals”, Acta Appl. Math., 162:1 (2019), 47–61
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
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
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
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
W. P. Hooper, B. Weiss, “Rel leaves of the Arnoux–Yoccoz surfaces”, Sel. Math.-New Ser., 24:2 (2018), 875–934
A. Ya. Maltsev, “The Second Boundaries of Stability Zones and the Angular Diagrams of Conductivity for Metals Having Complicated Fermi Surfaces”, J. Exp. Theor. Phys., 127:6 (2018), 1087
Citation in format AMSBIB
Contact us: