# American Institute of Mathematical Sciences

April  2016, 36(4): 1983-2025. doi: 10.3934/dcds.2016.36.1983

## A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes

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Received  November 2014 Revised  July 2015 Published  September 2015

Rauzy Classes and Extended Rauzy Classes are equivalence classes of permutations that arise when studying Interval Exchange Transformations. In 2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data from Translation Surfaces, which are associated to IET's thanks to the Zippered Rectangle Construction of Veech from 1982. In 2009, Boissy finalized the classification of Rauzy Classes also using information from Translation Surfaces. We present in this paper specialized moves in (Extended) Rauzy Classes that allow us to prove the sufficiency and necessity in the previous classification theorems. These results provide a complete, and purely combinatorial, proof of these known results. We end with some general statements about our constructed move.
Citation: Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1983-2025. doi: 10.3934/dcds.2016.36.1983
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