在一個微分方程的初始值問題中,它的解稱為一個半流(semiflow),且形成一個動力系統(Dynamical System)。在§1中,我們列出了一些主要的結果。在§2與§3中,我們最主要是在討論初始值問題解的擬保角性。在§2中,我們限制這個微分方程中的向量場(vector field)F(t,y),並且引進Ahlfors所定義的微分算子(differential operator)SF(見定義2.3),再要求的連續性、有界性與SF的有界性,則生成的半流會是唯一存在而且是擬保角的(quasiconformal),而且它的最大伸縮率(maximal dilatation)可用來估計(見定理2.4)。這主要是整理自Ahlfors[5]的工作。在§3中,我們對F的要求再放寬為只要求其局部Lebesgue可積(Lebesgue integral)與對幾乎所有的t,是連續函數並有局部廣義導數(見定義3.2),則此微分方程亦會生成唯一的一個以SF的模(norm)估計最大伸縮率的擬保角半流(見定理3.1)。這主要是整理自Sarvas[12]的工作。在§4中,我們任給定一個三維球體上的擬保角自同構(quasiconformal automorphism),我們將證明存在有一個向量場,使生成的動力系統所形成的同痕(isotopy,見定義4.1)連接它到恆等變換(見定理4.8)。這主要是整理自Seleznev[18] 的工作。而在§5中,我們將以不同的方式導出二維單位球面上同樣的問題也會成立,即我們任給定一個二維單位球面上的一個擬保角自同構,我們將要詳細說明如何造出一個向量場,使得生成的動力系統所形成的同痕連接它到恆等變換。而且我們將利用其性質來證明其生成的解是唯一存在的(見定理5.17)。而順便一提的是:我們定義了另一種較為放寬的函數類(見定理5.1)可以給定理4.8的另一種證明。這主要是整理自Seleznev[14]的工作。§4與§5兩節也讓我們了解決定三維球體與二維球面上擬保角變換的動力系統其特殊性質。

Given an initial value problem. Its solution is called a semiflow,and forms an dynamical system. In section 1,we list some main results aboutit. In section 2 and 3,we deal with the quasiconformality of the solution. In section 2,we restrict the vetor field of the differential equation,and introduce the differential operator defined by Ahlfors SF and ask the continuity,boundedness and the boundedness of SF,then the semiflowexists uniquely and is quasiconformal.Its maximal dilatation can be estimated.This comes of Ahlfors[5]. In section 3,we stretch the rule:ask F is Lebesgue integral only and for a.e. t,F is continuous and has local weak partial derivative,then this differential equation also generated a quasiconformal semiflowwhose maximal dilatation is estimated by the norm of SF.This comes of Sarvas[12]. In section 4,we give a quasiconformal automorphism of 3-dimensional ball.We will show that there exists a vector field generating an dynamicalsystem forming an isotopy joining it to identity.This comes of Seleznev[18].In section 5,we give a different way to show the same question on 2-dimensional sphere,i.e. given a quasiconformal automorphism of 2-dimensional sphere,we will explain with detail to how to construct a vector field generating a dynamical system forming an isotopy joiningit to identity. We will use its properties to prove the solution exists uniquely.By the way:we define another class of functions which give anotherproof of theorem 4.8.This comes of Seleznev[14]. Section 4 and 5 can make us understand the special property of dynamical system determing the quasiconformal mapping on the 3-dimensional ball and 2-dimensional sphere.