Self-Inverses, Lagrangian Permutations and Minimal Interval Exchange Transformations with Many Ergodic Measures

TL;DR

This paper proves the existence of self-inverse permutations in every Rauzy Class, shows they are Lagrangian, and constructs examples demonstrating the sharpness of ergodic measure bounds in minimal interval exchange transformations.

Contribution

It provides explicit constructions of self-inverse permutations in all Rauzy Classes and demonstrates their Lagrangian property, advancing understanding of interval exchange transformations.

Findings

Self-inverse permutations exist in every Rauzy Class.

Self-inverse permutations are Lagrangian in the homological sense.

Constructed examples show the bound on ergodic measures is sharp.

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. 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 G. Forni. W. A. 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 construcing examples in each Rauzy Class.