The Local Connectivity And Quasi-regular Extension Of Julia Sets

The thesis contains two main topics.One is to study the local connectedness of Julia sets of a class of holomorphic maps with bounded type Siegel disks.The other one is the quasiregular extension of certain class of one dimensional covering maps.Based on this extension,we construct a class of entire functions which may serve as the models for the Stallard's Cauchy integrals.As regards the first topic,we mainly study the local connectedness of the Julia sets of a class of holomorphic maps with bounded type Siegel disks.The earliest result in this aspect was due to Petersen[62].He proved that the Julia sets of quadratic polynomials with bounded type Siegel disks are locally connected.The fundamental ideas in Petersen's proof consist of a set of dynamically defined puzzle pieces and a delicate estimate of their geometry.With the aid of complex bounds for critical circle mappings,Yampolsky gave an alternative proof of Petersen's result.The gain of the complex bounds relies on the fact that there is a single critical point in the Siegel disk boundary.With the consideration of these,there are two new features in this work.The first one is that there may not exist a puzzle construction for the maps we study here,and the second one is that we allow the presence of more than one critical point in the Siegel disk boundary.Specifically,we study the locally connectedness of Julia sets of polynomial,rational maps and entire functions having a bounded type Siegel disk with quasi-circle boundary(Theory 4.1,p42).It is worth to point out that,when a Siegel disk contains an asymptotic value,there will be infinitely many pre-images of the Siegel disk with spherical diameter bounded away from zero.By a theorem of Whyburn,the Julia set is not locally connected in this case.The simplest such example is e2?i?zez.The study on the structure of the Julia set of e2?i?zez will be the second contents in our first topic(Theory 4.1,p44).The main tool we used in the proofs is a lemma which asserts that long iteration of certain Blaschke products exhibits weak expanding properties(Lemma 4.1,p46).As a complement,we discuss how to use quasiconformal surgery to construct a class of entire functions,such that they satisfy the following specific dynamic properties:all Fatou components are quasi disks,but their Julia sets are all non-locally connected(Lemma 5.1,Theory 5,1 p67).In[14],the authors there have constructed such kind of entire functions by Maclane-Vinberg method.Here we will use quasiconformal surgery to construct such functions.Compared with their method,which is purely functional theoretic,ours has more geometric feature.As regards the second topic,we study the quasiregular extension of certain class of one-dimensional covering maps from the real axis to the unit circle.The importance of the research lies in the following two facts.In the aspect of the theory of quasi-conformal mappings,it is a natural generalization of Beurling-Ahlfors' quas-conformal extension.In the aspect of the application in complex dynamics,it allows us to construct a quasi-regular model with an invariant line field,by which we may construct an entire function with certain desired dynamical properties through quasi-conformal surgery.In particular,we have constructed the models for Stallard's Cauchy integrals in the study of the Hausdorff dimension of the Julia sets of entire functions.In this part,we established sufficient conditions so that the one-dimensional covering map can be extended to a quasi-regular map(Theory 6.1,p81).The main idea is the con-struction of certain "Pyramid" models.As an application,we construct entire functions with the aid of quasiconformal surgery which may serves as models for Stallard's Cauchy integrals.The structure of the article is as follows.In Chapter 1,we first introduce the history and the most recent developments with regard to the two topics discussed in the thesis.We then present the relative background materials in complex dynamics including hyperbolic metric,Koebe's distortion theorem,Hausdorff dimension and critical circle homeomorphism maps.In Chapter 2,we introduce the theory of quasiconformal mappings comprehensive-ly.In particular,we focus our attention on the quasiconformal extension of the given boundary maps.In Chapter 3,we pay more attention to some basic principles of the quasiconformal surgery and its application in the complex dynamic system.We first introduce the basic principles of quasiconformal surgery and their equivalent conditions used in Chapters 4 and 5.In addition,we present some applications of the quasiconformal surgery in the complex dynamics.For example,how polynomial-like mappings are related to actual polynomials is explained by quasiconformal surgery,how to convert an attracting point into a super-attracting point by quasiconformal surgery and how to turn Siegel disks into Herman rings and so on.In Chapter 4,we divide the chapter into three parts including seven subsections to introduce our main works.In more detail,we prove that a long iteration of a class of quasi-Blaschke models has certain expanding property near the unit circle.This leads us to prove the local connectivity of the Julia sets of a number of holomorphic maps with bounded type Siegel disks.In Chapter 5,we use the quasiconformal surgery to construct entire functions with all the Fatou components being quasi disks and Julia sets being non-locally connected.The key point of the construction is to find a quasi-regular mapping F which serves as the pre-model for the desired entire function f.In Chapter 6,we study the quasiregular extension of certain class of one-dimensional covering mappings.In the theorem 6.1,we give the sufficient conditions that the covering maps from the real axis to the unit circle can be quasi-regularly extended to the upper half plane.Our method of the proof is constructive.The main idea is the construction of Pyramid models.As an application,we construct entire functions which may serve as the models for the Stallard's Cauchy integrals.