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

Jon  Fickenscher

doi:10.3934/dcds.2016.36.1983

Article Contents

2016, Volume 36, Issue 4: 1983-2025. Doi: 10.3934/dcds.2016.36.1983

This issue Previous Article Real bounds and Lyapunov exponents Next Article The three-dimensional center problem for the zero-Hopf singularity

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

Jon Fickenscher^1,

1.
Fine Hall - Washington Road, Princeton, NJ 08544-1000

Received: October 31, 2014

Revised: June 30, 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 37A05; Secondary: 30F30, 32G15, 37E35.

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References

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