[Paper Review] Self-inverses in Rauzy Classes

Jonathan Michael Fickenscher|arXiv (Cornell University)|Mar 17, 2011

Mathematical Dynamics and Fractals20 references3 citations

TL;DR

This paper proves that every Rauzy class of interval exchange transformations contains at least one self-inverse permutation, providing an explicit construction method using symmetric blocks. It further establishes necessary and sufficient conditions for generalized Rauzy classes to contain self-inverse generalized permutations, and demonstrates that self-inverse permutations are Lagrangian, offering a new proof of Forni's lemma on periodic vertical trajectories in abelian differentials.

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.

Motivation & Objective

• To prove that every true Rauzy class contains at least one self-inverse permutation.
• To provide an explicit construction of such self-inverse permutations using symmetric block patterns.
• To extend the result to generalized Rauzy classes by identifying necessary and sufficient conditions for the existence of self-inverse generalized permutations.
• To establish a connection between self-inverse permutations and Lagrangian subspaces in the homology of flat surfaces.
• To offer an alternative proof of Forni’s Lemma 4.4 on the dimension of the space spanned by periodic vertical trajectories in abelian differentials.

Proposed method

• Constructs self-inverse permutations via symmetric concatenation of blocks that preserve the required signature and type of the Rauzy class.
• Uses the fact that Rauzy induction conjugates a permutation with its inverse, implying closure under inversion.
• Applies the classification of Rauzy classes by signature and type (Corollary 1.22) to verify membership of constructed permutations.
• For generalized permutations, introduces block insertion techniques to control the signature and satisfy parity conditions on singularity degrees.
• Defines Lagrangian permutations as those for which vertical trajectories span a g-dimensional subspace in homology, and proves self-inverse permutations are Lagrangian.
• Employs symplectic forms and linear algebra to verify linear independence of trajectory vectors, confirming the Lagrangian property.

Experimental results

Research questions

• RQ1 Does every Rauzy class contain a self-inverse permutation?
• RQ2 What conditions must a generalized Rauzy class satisfy to contain a self-inverse generalized permutation?
• RQ3 How are self-inverse permutations related to Lagrangian subspaces in the homology of flat surfaces?
• RQ4 Can the existence of self-inverse elements be used to reprove known results about periodic vertical trajectories in abelian differentials?
• RQ5 What is the role of block structure in constructing self-inverse permutations that preserve Rauzy class invariants?

Key findings

• Every true Rauzy class contains at least one self-inverse permutation, as proven by explicit construction using symmetric blocks.
• A generalized Rauzy class contains a self-inverse generalized permutation if and only if the number of singularities of each odd degree (≥ -1) is even, and the total number of singularities of degree 0 is even.
• Self-inverse permutations are Lagrangian, meaning the vertical trajectories of their suspensions span a g-dimensional subspace in homology.
• The permutation π = (d, d−1, ..., 2, 1) is transposition Lagrangian, as the associated trajectory vectors are linearly independent.
• Block-constructed self-inverse permutations are transposition Lagrangian, with linear independence of trajectory vectors verified via symplectic pairing arguments.
• The result provides an alternative proof of Forni’s Lemma 4.4, showing that the space spanned by periodic vertical trajectories has dimension g in the moduli space of abelian differentials.