论文标题
在更高维度的Lyapunov-Poincaré定理上
On a theorem of Lyapunov-Poincaré in higher dimension
论文作者
论文摘要
如果单一的第一架单射流是非重新等级的，那么经典的Lyapunov-Poincaré中心定理定理就可以确保在中心奇异性附近的中心奇异性附近的第一个积分组成。基本点是存在给定的单一形式的分析第一积分。在本文中，我们考虑了两个主要框架的概括：（i）在较高维度和（ii）维度二维中奇异的圆锥形叶子中的Codimension ONE的实际分析叶子。所有这些都与查找标准的问题有关，以确保具有合适的第一射流的给定的codimension One细菌的分析第一积分。我们的方法包括用全态叶子对中心定理进行解释，并在穆苏的想法之后，将全体形态叶子应用于获得所需的第一个积分。结果，我们能够对确切均匀的一种形式的可整合扰动进行一些REEB经典结果，并证明了这些结果的某些版本，这些版本是非异化的（横向摩尔斯类型）奇点的框架的框架。
The classical Lyapunov-Poincaré center theorem assures the existence of a first integral for an analytic one-form near a center singularity in dimension two, provided that the first jet of the one-form is nondegenerate. The basic point is the existence of an analytic first integral for the given one-form. In this paper we consider generalizations for two main frameworks: (i) real analytic foliations of codimension one in higher dimension and (ii) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal in the obtaining the required first integral. As a consequence we are able to revisit some of Reeb classical results on integrable perturbations of exact homogeneous one-forms, and prove some versions of these to the framework of non-isolated (perturbations of transversely Morse type) singularities.
