Harmonic Measure Versus Hausdorff Measure on Uniformly Perfect Sets; Geometric Properties of Julia Sets

摘要：本课题的目的是建立调和测度和Hausdorff测度在完全不连通的一致完备集（实线的子集或有理函数的julia集）上的奇异性。结果依赖于平面域边界附近的对数容量、格林函数和调和函数的估计。它将潜在的理论论证与遍历理论和动力系统结合起来，并使用了Carleson、Jones和Wolff、Makarov和Bourgain介绍的技术。我将要研究的分形集代表了一个动力系统的混沌部分。生命中几乎所有的现象都是动态系统：复杂的如天气、地震和疾病传播，简单的如台球游戏。调和函数和势理论首先引起了物理学家和工程师的兴趣，特别是在热力学和电磁学领域。今天所有成功的海浪都源于这个理论。天线间电磁信号的传输、天线的设计及其阵列布置是目前应用电势理论的问题之一。

ABSTRACT:The goal of the project is to establish the singularity of harmonic measure and of Hausdorff measure on some totally disconnected,uniformly perfect sets (subsets of the real line orJulia sets of rational functions). The results rely on estimates of logarithmic capacities, Green's function and harmonic functions near the boundary of plane domains. It combines potential theoretic arguments with ergodic theory and dynamical systems,and uses techniques introduced by Carleson, Jones and Wolff, Makarov and Bourgain.The fractal sets I will study represent the chaotic part of a dynamical system. Almost all the phenomenon in life are dynamical systems: complex ones like weather, earthquakes and spreading of disease, and simple ones, like the game of billiard.Harmonic functions and potential theory were first of interest to physicists and engineers, particularly in the fields of thermodynamics and electro-magnetics. All the successes in waves today stem from this theory. The transmission of signals electro-magnetically between antennas,the design of antennas and their placement in arrays is one of the current problems that apply potential theory.