Some Results And Questions On Quasisymmetric Mappings

In the present paper we summarize some results and questions on quasisymmetric mappings since 1980.Some examples and questions belong to the author.It consists of six parts.In Part 1,we recall the definition and some basic properties of quasisymmetric mappings.The quasisymmetry is difficult to be checked.We introduce weak quasisym-metric mappings and show that the weak quasisymmetry implies the quasisymmetry for many commonly used metric spaces.In Part 2,we give a condition that enables a subfamily of quasisymmetric map-pings to be normal.In Part 3,we discuss the properties of power quasisymmetric mappings and prove that a quasisymmetric mapping between two uniformly perfect spaces is power quasi-symmetric.In part 4,we prove that a metric space is quasisymmetically equivalent to an Ahlfors regular space if and only if it is uniformly perfect and carries a doubling mea-sure.In Part 5,we show that the quasisymmetric characteristic properties of the middle-third Cantor set are compact,doubling,uniformly perfect,and uniformly disconnected.In other words,every metric space with these four properties is quasisymmetrically equivalent to the middle-third Cantor set.In Part 6,we discuss the relationship between quasisymmetric mappings and qua-siconformal mappings.Generally,quasisymmetric mappings are quasiconformal,but not conversely.We give some situations where quasiconformal mappings are quasisym-metric.