[[alternative]]Dynamical Systems Determing Quasiconformal Mappings

[[abstract]]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.