李代数的模演算正规化及其应用
Normalization in Lie algebras via mould calculus and applications
摘要：
我们建立了一个抽象的李理论环境中的Ecalle的模具演算，并使用它来解决规范化问题，它涵盖了动力系统理论中的几个形式规范形问题。模形式主义使我们能够减少李理论问题的模具方程，其中的解决方案是非常明确的，可以完全描述的规范变换群。动力学的应用包括构造围绕平衡点的向量场的Poincar{é}-Dulac形式正规形，构造具有快速角变量（Hamilton或非Hamilton）的向量场的形式无限阶多相平均过程，或者构造经典和量子情形下的Birkhoff正规形.作为副产品，我们得到，在谐振子的情况下，量子Birkhoff形式的经典收敛，没有任何丢番图假设的频率上的未扰动的哈密顿。
英文摘要：
We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincar{é}-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.
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