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

一致完美集上的调和测度与 Hausdorff 测度；

• 批准号: 0070769
• 负责人: Irina Popovici
• 金额: $ 7.89万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070769&HistoricalAwards=false
• 关键词: Harmonic; Measure; Versus; Hausdorff; Uniformly

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.

摘要：本项目的目标是建立调和测度和Hausdorff测度在完全不连通的一致完备集（有理函数的真实的直线或Julia集的子集）上的奇异性。这些结果依赖于平面区域边界附近的对数容量、绿色函数和调和函数的估计。它结合了潜在的理论论点与遍历理论和动力系统，并使用Carleson，Jones和Wolff，Makarov和Bourgain介绍的技术。生命中几乎所有的现象都是动力系统：复杂的系统，如天气、地震和 调和函数和势能理论首先引起物理学家和工程师的兴趣，特别是在热力学和电磁学领域。今天所有的成功浪潮都源于这一理论。信号在天线之间的电磁传输、天线的设计及其在阵列中的布置是应用势场理论的当前问题之一。