Linearization Of Rational Functions With Two-parameter Perturbation

The study of locally properties near periodic points of rational func-tion is one of major subject of complex dynamics. It is known that attracting and super-attracting periodic points of rational function belong to its Fatou set. Howev-er, repelling and rational neutral periodic points belong to its Julia set. Whether or not irrational neutral periodic points belong to its Fatou set? This comes down to lineariza-tion. Brjuno obtained that the rational function R is linearizable at irrational neutral periodic points, if the rotation a is a Brjuno number. Douady made a conjecture that for any rational function with degree more than one, if a does not meet the Brjuno condition, then R is not linearizable. Yoccoz showed that for quadratic polynomials the Douady conjecture is true. So far, it is still open for cubic polynomials. Geyer obtained that the conjecture is true for certain family of polynomials with some spe-cial properties. In this thesis, using complex analysis of several variables and idea of Geyer, and making a perturbation of two-parameter, a sufficient condition for Douady conjecture is obtained.