Research On Measurable And Topological Dynamics Of Transcendental Meromorphic Functions

In this dissertation, we study the measurable and topological dynamics of tran-scendental meromorphic functions. The measurable dynamics focus on the existenceand geometric properties of the conformal measures for two types of meromorphicfunctions. The topological dynamics are mainly related to the functions of class Kand the meromorphic functions with at least one direct tract. We investigate Baker do-mains, wandering domains, Julia sets of special forms, Julia sets with positive measureand the topological structure of the escaping set.This thesis consists of six chapters. The first chapter is about the background andthe basic knowledge. We will give out the necessary knowledge on complex analysisand introduce the theory of complex dynamic system.In Chapter2, we investigate the problem of the existence of conformal measuresfor some transcendental meromorphic functions. Firstly, we discuss the atom and non-atom conformal measures of parabolic meromorphic functions on the Riemann sphere,and we give a lower bound of the Julia set of these functions. With some growth condi-tion about the derivative, we prove the existence of conformal measures for parabolicmeromorphic functions with finite order. By estimating the function mt(Bσ(z, r)) forthe radial point z, we prove that any two eP(t)|f′|σt-conformal measures are equivalentrestricted on Jr(f). For the zero h of the pressure function, we get the relationship be-tween the h dimension Hausdorf measure and the conformal measure, then we obtainan upper bound for dimH(Jr(f)).In Chapter3, we work on the topology dynamic system of functions of class K.Adopting the method of logarithmic change, we prove the existence of some specialBaker domains starting from the Siegel disk, attracting domain, parabolic domain andHerman ring, respectively. We also give out a function from class K which has somespecial wandering domains. Furthermore, we discuss the functions with Julia sets ofspecial type. In the end, we take into account the Julia sets with positive measure or being complex plane.In Chapter4, we study the topology dynamic system of functions with at least onedirect tract: we discuss the topology structure of the escaping points contained in thesame tract and prove some results about fast escaping sets by using the technique oflevel sets. We also give a meromorphic function with slow escaping set.Chapter5summarizes the results of the whole thesis.In Chapter6, we raise some questions which are worth investigating in the future.