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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 

 

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
DOI: https://doi.org/10.1090/mosc/246
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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Additional 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

DOI: https://doi.org/10.1090/mosc/246
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