Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many Ergodic measures

Thanks to works by Kontsevich and Zorich followed by 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. We will also show that self-inverse permutations are Lagrangian, meaning any suspension has its vertical cycles span a Lagrangian subspace in homology. This will simplify the proof of a lemma in a work by Forni. Veech proved a bound on the number of distinct ergodic probability measures for a given minimal interval exchange transformation. We verify that this bound is sharp by constructing examples in each Rauzy Class.