On Normal Families Of Quasisymmetric Mappings And Their Applications

The normal family theory of analytic functions plays an important role in the proof of Riemann mapping theorem.At the same time,it is the basis of complex dynamical system theory.In this paper,we discuss the normal families of quasisymmetric mappings and their applications in the study of quasisymmetric equivalence.Based on some known results,we give some sufficient conditions for a family of quasisymmetric mappings to be normal.In addition,we give some methods for constructing normal families of quasisymmetric mappings.The normality of quasisymmetric mapping sequences can be used to study the quasisymmetric equivalence of spaces.Often,a quasisymmetric equivalence problem can be reduced to constructing a sequence of quasisymmetric mappings and proving that the sequence of quasisymmetric mappings is normal,so the limit function is the desired quasisymmetric mapping.Two examples are given to illustrate this.This paper is organized as follows:In the first part of this paper,we review the normal family theory of analytic func-tions and its applications.In the second part,we summarize some basic properties of quasisymmetric map-pings.Especially,we recall some results on the relationship between quasisymmetric and quasiconformal mappings and quasisymmetric equivalence of spaces.The third part is the main content of the research.We give some sufficient con-ditions for a family of quasisymmetric mappings to be normal,and then give some methods for constructing the normal families of quasisymmetric mappings.Finally,we give two examples to show that the normal families of quasisymmetric mappings can be applied to solve some quasisymmetric equivalence questions of spaces.