The Linearization Of The Irrationally Neutral Periodic Points Of Analytic Functions

Complex dynamics is one of branches of complex analysis,which wasfounded by Fatou and Julia in the 1920s. At that time, one of the main mo-tives was to discuss some functional equations through iterates, further to studytheir dynamics. Rich connecting with other domains, this subject has arousedwidespread attention and has become one of the most active branches in math-ematics. In fact, Yoccoz and McMullen, which obtained Fields prize separatelyin 1994 and 1998, were engaged in the study of complex dynamics.Dynamics on rational functions is the most systematic and comprehensive.A series of e?ective methods has been established. But dynamics on irrationallyneutral periodic points and their neighborhoods is not clear. It has been foundthat irrationally neutral periodic points of the analytic function belong to itsFatou set if and only if the analytic function is conformally conjugate to itslinear part at the irrationally neutral periodic points and their neighborhoods.Therefore,it is very essential and meaningful to discuss the linearization of theanalytic function at the irrationally neutral periodic points. A su?cient con-dition on it, which is called Brjuno condition([definition 2.1.2]), is obtained bymathematicians, Yoccoz, Siegel, Brjuno and Ru¨ssmann, and further the condi-tion is the best for quadratic polynomials. However, we don't know whether thiscondition is also the best for general analytic functions. Here we will consider this problem.This dissertation divides into two chapters. The first chapter introducesrelevant background knowledge, including Riemann surfaces, polynomial-likemappings, holomorphic motions and so on.In the second chapter, using the theory of small divisors we study thelinearization of analytic function at the irrationally neutral periodic points.Because a analytic function and its iterates have the same Fatou set, it issu?cient to consider the linearization of the analytic function near a fixed point.If the analytic function is linearizable near a fixed point, then the fixes point isin its Fatou set and the connected component which contains the fixed point is atopology disk, called a Siegel disk. The restriction of the function in this Siegeldisk is univalent. Therefore, we consider a more general setting: a function f(z)is univalent and analytic, and f(0) = 0,f (0) = e^2πθi (θ∈R \ Q). Applyingpolynomial-like mappings and holomorphic motion theory and a technique ofX.Bu and A.Ch′eritat[24], we obtain that if the function Ga, A(z) = f(z) +a z2 (0 < |a|≤1) is linearizable, let fa, A(z) = a^-1f(az) + Az²,then thefunction log R(fa, A) is harmonic, where R(fa,A) is the convergent radius of thelinearizing formal power series of fa,A, and so the function f(z) satisfies Brjunocondition. At the same time, the average of log R(Ga, A) at some circle is lessthan log R(f).