The Loewner Differential Equation And Cauchy Transform Of Some Self-similar Measures

This article consists of five chapters. In the first half of this article, we study some properties of the Loewner differential equation in the upper half-plane. In the second half of the article, we study the properties of the Taylor coefficients of Cauchy transforms of some self-similar measures and the prop-erties of the holomorphic proper maps in the complex plane.In Chapter1, the background, motivation and current situation of the Loewner differential equation and Cauchy transforms of some self-similar mea-sures are introduced. At the same time, the main results of this article are given.In Chapter2, we mainly consider the chordal Loewner differential equa-tion in the upper half-plane, the behavior of the driving function λ and the generated hull Kt when Kt approaches λ(0) in a fixed direction or in a sector. In the case that the hull Kt is generated by a simple curve γ(t) with γ(0)=0, we prove some sharp relations of λ(t)/(?)t and γ(t)/(?)t as t tends to0which improve the previous work of other people. Under the assumption that the hull Kt is generated by the curve γ(t), we study the self-intersection of the curve γ(t) in the finite time and give a sufficient condition and a necessary condition.In Chapter3, we discuss the multiple-slit version of Loewner differential equation in the upper half-plane. Under the assumption that the hull Kt consists of some disjoint slits, we give the relationship between the driving functions λ1(t), λ2(t),…,λn(t) and the multiple-slit γi(t),γ2(t),…,γn(t) as t tends to0.In Chapter4, we consider the Taylor coefficients{bn}∞n=1of Cauchy trans-forms F(z) of some self-similar measures and proved that{naRnbn}∞n=1is dense in some non-degenerated bounded interval, where a is the Hausdorff dimension of self-similar set K and R is the maximum radius of analytic region. In this chapter, we also give the way to divide the domain into univalent regions for the holomorphic proper maps. In Chapter5, we give some questions which have close relationship to this paper for the reader.