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 conjecture analogous to that of Marmi–Moussa–Yoccoz, stated for almost all IETs, in the special case of self-similar IETs.

在本文中，我们使用重归一化方法研究广义区间交换图的局部刚度属性。我们研究了重归一化算子\（\ mathcal {R} \）在\（\ mathcal {C} ^ {3} \）的空间上的动力学-固定点上的广义区间交换变换（这是标准周期类型IET） ）。我们证明\（\ mathcal {R} \）是双曲线的，并且不稳定方向的数目恰好是由IET的遍历理论以及Forni和Marmi–Moussa–Yoccoz的工作所预测的。结果，我们证明了具有d个间隔的周期间隔交换变换的局部\（\ mathcal {C} ^ 1 \）-共轭类，其关联表面具有属g及其Lyapounoff指数均非零的是一个余维\（g-1 + d-1 \） \（\ mathcal {C} ^ 1 \） - \（\ mathcal {C} ^ {3 } \）-广义间隔交换转换。这解决了类似于Marmi-Moussa-Yoccoz的猜想，在自相似IET的特殊情况下，几乎针对所有IET都提出了这样的猜想。