The Application Of The Quasiconformal Function Theory To The Linearizable Problems

This paper studies problem of linearizability of analytic function by meansof quasiconformal. Some results of Geyer are extended. The first part of thispaper introduces some basic theories including iteration of rational functions,linearization of analytic function and quasiconformal function. In the secondpart of this paper, we shall prove Main Theorem. In 1971, Brjuno has provedthat f(z) = e2πiθz + O(z2) is available linearized whenθis Brjuno number(definition 1.2.3). Furthermore, Yoccoz obtained that the Brjuno conditionis the best for quadratic polynomials Pθ= e2πiθz + z2 in 1995. However, wedon't know whether this condition is also the best for general analytic functions.This issue has attracted the attention of many mathematicians, such as Yoccoz,Douady, and so on. They have carried out thorough research on this problem.In this paper, using the theory of quasiconformal we will show that if analyticfunction f(z) plusing a proper secondary disturbance is available linearized,then f(z) + Az2 and Pθare conformal conjugated, thusθ∈B.