具有奇异底层几何结构的动力系统的局部normal forms
Local normal forms of dynamical systems with a singular underlying geometric structure
摘要：
在本文中，我们证明了一个同时存在的局部正规化的夫妇$(X,\mathcal{G})$，其中$X$是一个向量场，消失在一个点和$\mathcal{G}$是一个奇异的潜在的几何结构是不变的$X$，在许多不同的情况下：奇异体积形式，奇异辛和泊松结构，和奇异切触结构。类似于哈密顿向量场的Birkhoff正规化，我们的正规化也只是形式上的。然而，当$\mathcal{G}$和$X$是（实的或复的）解析的，并且$X$是解析可积的或达布可积的，那么我们的同时归一化也是解析的。我们的证明是基于复曲面的方法规范化的动力系统，复曲面的守恒律，和等变路径方法。我们还考虑了当$\mathcal{G}$是奇异的但$X$在原点不为零的情况。
英文摘要：
In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $\mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $\mathcal{G}$ is singular but $X$ does not vanish at the origin.
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