Uniformization Of Quasicircles Carpets

A set T(?)C is a quasicircle capet if and only if int(T)=(?)and it can be written as(?) where {Di} is a collection of pairwise disjoint closed Jordan regions and(?)Di are qua-sicircles.The uniformization question is when a quasicircle carpets can be mapped by a quasicoformal mapping onto a carpet whose peripheral circles are all geometric cir-cles.In the thesis,we discuss some conditions under which a quasicircle carpet can be uniformizated as a round carpet.The contents are as follows:In Chapter 1,we make an overview of the background and significance of the article topics,and also present the main notation of this paper.In Chapter 2,we recall definitions and basic properties of quasiconformal,qua-sisymmetry and quasi-Mobius.Especially,we show that these classes of mappings can be converted to each other under certain conditions.In Chapter 3,we discuss the definition of quasicircles and its equivalence.Then we recall uniform quasicircle carpets and carpets with the uniformly relatively separated property.we prove that these two classes of carpets are preserved under quasi-Mobius mappings.In Chapter 4,we prove that every quasi-Mobius embedding from T into C has a quasiconformal extension.In Chapter 5,we give some bounds estimate of the transboundary modulus by calculation.Finally,we give a proof of this paper and show some counterexample.