Linearization And The Locally Dynamical Property Of Cremer Points

Complex dynamics is one of branches of complex analysis, which wasfounded by Fatou and Julia in the 1920s. Since rich connecting with otherdomains, this subject has aroused widespread attention. More and more math-ematicians have worked at this. Complex dynamics has become gradually oneof the most active branches in mathematics.Researching for complex dynamics mainly concentrated on rational func-tions, entire functions, holomorphic mappings on C⁺ (C⁺ = C - {0}, C is acomplex plain), and transcendental meromorphic functions . Dynamics on ra-tional functions is the most systematic and comprehensive. A sery of wonderfulresults has been established. But there are many problems which haven't beanswered . For example, since irrationally neutral periodic points may be inFatou set or Julia set, and dynamics on irrationally neutral periodic points andtheir neighborhoods is very complex , its property is not clear. It has been foundthat irrationally neutral periodic points of analytic function belong to its Fatouset if and only if the analytic function is locally linearizable at the irrationallyneutral periodic points and their neighborhoods. Therefore, it is very essentialand meaningful to discuss the linearization of the analytic function at the ir-rationally neutral periodic points. Yoccoz, Brjuno, and many mathematicians have done large amount of works, and have obtained many important results.There is a suffcient condition on it, which is called Brjuno condition(definition2.1.2), and Yoccoz has showed that the condition is optimal for quadratic poly-nomials. However, we don't know whether this condition is also optimal forgeneral analytic functions. Douady has proposed the biggest conjecture aboutthe linearizable problem in [2]: If P(z) = e^2Ï€iαz+a₂z²+···+a_dz^d, ad = 0, d≥2is a linearizable polynomial withα∈R, thenα∈B. In process of exploringthis question , this article obtains a property about the Cremer point.This dissertation is divided into two chapters. The first chapter introducesrelevant background knowledge, including Riemann surfaces, complex dynamicsof rational functions and so on.In the second chapter, we will discuss the linearization of analytic functionat the irrationally indifferent periodic points and their neighborhoods. Considera more general situation: a function f(z) is univalent and analytic on the unitdisk, and f(0) = 0,f (0) = e2παi (α∈R\Q). In this paper, we give two ideas toanswer the linearizable problem of Douady and obtain a local property aboutthe Cremer point.