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Local rigidity for periodic generalised interval exchange transformations
Inventiones mathematicae ( IF 2.6 ) Pub Date : 2021-05-27 , DOI: 10.1007/s00222-021-01051-3
Selim Ghazouani

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.



中文翻译:

周期广义区间交换变换的局部刚性

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\(\ mathcal {R} \)\(\ mathcal {C} ^ {3} \)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。

更新日期:2021-05-27
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