Self-Inverses in Rauzy Classes
Author
Fickenscher, Jonathan Michael
Date
2011Advisor
Veech, William A.
Degree
Doctor of Philosophy
Abstract
Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have
a classification of all Rauzy Classes of any given genus. It follows from these works
that Rauzy Classes are closed under the operation of inverting the permutation. In
this paper, we shall prove the existence of self-inverse permutations in every Rauzy
Class by giving an explicit construction of such an element satisfying the sufficient
conditions. As a corollary, we will give another proof that every Rauzy Class is closed
under taking inverses. In the case of generalized permutations, generalized Rauzy
Classes have been classified by works of M. Kontsevich, H. Masur and J. Smillie, E.
Lanneau, and again C. Boissy. We state the definition of self-inverse for generalized
permutations and prove a necessary and sufficient condition for a generalized Rauzy
Class to contain self-inverse elements.
Keyword
Mathematics