# American Institute of Mathematical Sciences

February  2012, 32(2): 643-656. doi: 10.3934/dcds.2012.32.643

## Symmetric interval identification systems of order three

 1 Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russian Federation

Received  November 2010 Revised  June 2011 Published  September 2011

In the present paper we study symmetric interval identification systems of order three. We prove that the Rauzy induction preserves symmetry: for any symmetric interval identification system of order 3 after finitely many iterations of the Rauzy induction we always obtain a symmetric system. We also provide an example of symmetric interval identification system of thin type.
Citation: Alexandra Skripchenko. Symmetric interval identification systems of order three. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 643-656. doi: 10.3934/dcds.2012.32.643
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##### References:
 [1] M. Bestvina and M. Feighn, Stable actions of groups on real trees, Invent. Math., 121 (1995), 287-321. [2] M. Bestvina, $\mathbbR$-trees in topology, geometry, and group theory, in "Handbook of Geometric Topology," North-Holland, Amsterdam, (2002), 55-91. [3] M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory and Dynamical Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652. [4] H. Bruin and S. Troubetzkoy, The Gauss Map on a class of interval translation mappings, Israel J. Math, 137 (2003), 125-148. doi: 10.1007/BF02785958. [5] I. Dynnikov, Interval identification systems and plane sections of 3-periodic surfaces, Proceedings of the Steklov Institute of Mathematics, 263 (2008), 65-77. doi: 10.1134/S0081543808040068. [6] I. Dynnikov and B. Wiest, On the complexity of braids, J. Eur. Math. Soc., 9 (2007), 801-840. doi: 10.4171/JEMS/98. [7] I. Dynnikov, Semiclassical motion of the electron. A proof of the Novikov conjecture in general position and counterexamples, in "Solitons, Geometry, and Topology: On the Crossroad" AMS Transl., Ser. 2, 179, Amer. Math. Soc., Providence, RI, (1997), 45-73. [8] D. Gaboriau, Dynamique des systèmes d'isométries: Sur les bouts des orbits, Invent. Math., 126 (1996), 297-318. doi: 10.1007/s002220050101. [9] G. Levitt, La dynamique des pseudogroupes de rotations, Invent. Math., 113 (1993), 633-670. doi: 10.1007/BF01244321. [10] S. P. Novikov, The Hamiltonian formalism and many-valued analogue of Morse theory, Usp. Mat. Nauk, 37 (1982), 3-49.
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