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.
Categories: math.DS