# Local rigidity for periodic generalised interval exchange transformations

## TL;DR

This paper investigates the local rigidity of generalized interval exchange transformations using renormalisation, proving hyperbolicity of the renormalisation operator and characterizing the conjugacy class as a submanifold, thus confirming a conjecture.

## Contribution

It establishes the hyperbolic nature of the renormalisation operator and describes the local conjugacy class structure, solving a specific case of a conjecture by Marmi-Moussa-Yoccoz.

## Key findings

- Renormalisation operator $\\mathcal{R}$ is hyperbolic.
- Number of unstable directions matches predictions.
- Local conjugacy class forms a codimension $g-1 + d-1$ submanifold.

## Abstract

In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operator $\mathcal{R}$ acting on the space of $\mathcal{C}^{3}$-generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show that $\mathcal{R}$ is hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi-Moussa-Yoccoz. As a consequence we prove that the local $\mathcal{C}^1$-conjugacy class of a periodic interval exchange transformation, with $d$ intervals, whose associated surface has genus $g$ and whose Lyapounoff exponents are all non zero is a codimension $g-1 +d-1$ $\mathcal{C}^1$-submanifold of the space of $\mathcal{C}^{3}$-generalised interval exchange transformations. This solves a particular case of a conjecture of Marmi-Moussa-Yoccoz.