[论文解读] Introduction to 1-summability and resurgence

David Sauzin|arXiv (Cornell University)|May 2, 2014

Advanced Differential Equations and Dynamical Systems参考文献 23被引用 40

一句话总结

本文提供了对1-可求和性及Écalle的重生理论的自包含介绍，重点在于发散形式幂级数的Borel-Laplace求和。它证明了重生级数在复合运算下构成封闭代数，并将该理论应用于分类抛物型情形下的切于恒等映射的全纯微分同胚芽，以及通过奇异微积分和Stokes现象分析非线性微分方程的解。

ABSTRACT

This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series.In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples are given a thorough treatment (the Euler series, the Stirling series, a less known example by Poincaré). Special attention is devoted to non-linear operations: 1-summable series as well as resurgent series are shown to form algebras which are stable by composition. As an application, the resurgent approach to the classification of tangent-to-identity germs of holomorphic diffeomorphisms in the simplest case is included. An example of a class of non-linear differential equations giving rise to resurgent solutions is also presented. The exposition is as self-contained as can be, requiring only some familiarity with holomorphic functions of one complex variable.

研究动机与目标

• 为复平面上某一方向弧段中1-可求和形式幂级数的Borel-Laplace求和建立严格的框架。
• 介绍Écalle的重生理论，强调Borel变换的解析延拓及其奇点的作用。
• 证明1-可求和级数与重生级数在非线性运算（如复合）下保持封闭性。
• 将重生理论应用于分类抛物情形下切于恒等映射的全holomorphic微分同胚芽。
• 利用奇异算子与符号Stokes自同态分析非线性微分方程解的Stokes现象与渐近行为。

提出的方法

• 利用复平面上某一方向的正式Borel变换与Laplace变换定义1-可求和性。
• 将Borel-Laplace和定义为在Gevrey意义下渐近于发散形式级数的全纯函数。
• 应用奇异微积分分析Borel变换的奇点，并通过奇异算子Δω关联不同的Borel-Laplace和。
• 使用符号Stokes自同态描述渐近展开在Stokes线两侧的不连续跃迁。
• 构建桥方程以关联符号Stokes自同态的作用与形式微分同胚的复合。
• 证明重生级数在形式微分同胚群下的复合与逆运算下保持封闭。

实验结果

研究问题

• RQ1 如何通过Borel-Laplace求和为收敛半径为零的发散形式幂级数赋予有意义的全纯函数？
• RQ2 在给定扇形内，何种条件可保证形式级数为1-可求和？不同求和方向之间有何关系？
• RQ3 奇异算子Δω如何捕捉渐近展开在Stokes线两侧的不连续行为？
• RQ4 重生理论以何种方式实现对切于恒等映射的全纯微分同胚芽的分类？
• RQ5 非线性微分方程如何产生重生解？符号Stokes自同态在其分析中扮演何种角色？

主要发现

• 1-可求和形式级数的Borel-Laplace和在某一扇形内存在全纯函数表示，且在Gevrey意义下渐近于该级数。
• 重生形式级数构成一个在复合运算下封闭的代数，使得可通过形式幂级数研究非线性动力系统。
• 符号Stokes自同态作用于桥方程的解空间，编码了渐近展开在Stokes线两侧的不连续变化。
• 对于一个具有零共振器的简单抛物型胚，其迭代函数是重生的，Fatou坐标通过horn映射与解析分类相关联。
• 奇异算子Δω在重生函数空间上作用为微分算子，其对桥方程解u*的作用由Δωu* = u* ∘ h*^up 或 h*^low 给出，具体取决于符号与方向。
• 奇异算子Δω的指数作用于u*等价于与形式微分同胚h*^up或h*^low的复合，从而在重生理论与微分同胚分类之间建立了深刻联系。