Mathematical Sciences: Harmonic measure and fractal sets

数学科学：调和测度和分形集

• 批准号: 9402946
• 负责人: Nikolai Makarov
• 金额: $ 15万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9402946&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; measure; fractal

9402946 Makarov This project emphasizes mathematical research on a basic concept in mathematical analysis known as harmonic measure. There are several equivalent definitions ranging from the probability that a Brownian path hits a fixed set to the equilibrium distribution of that set as understood in potential theory. In recent years there has been considerable interest in the study of global metric properties and characteristics of harmonic measure such as the dimension of the minimal Borel support, the size of exceptional sets for the boundary behavior of the Green function, the bounds for the dimension distortion under conformal maps, etc. At the same time close connections with the ergodic theory of analytic dynamical systems have been established. Harmonic measure has a natural characterization in dynamical terms for boundaries which are like Cantor sets or Julia sets, and ergodic theory provides a powerful tool for the study of its metric properties. In the opposite direction, the estimates of harmonic measure describe the geometry of the fractals and certain properties of the dynamics. The main goal of this project is to explore the fine structure of harmonic measure for general plane domains as well as for some important particular classes of fractals. A continuation of studies on various multifractal spectra that characterize the global metric properties of harmonic measure. Application to the ergodic theory of Julia sets deal with the problem of analyticity of the corresponding pressure functions and with the phase transition phenomenon. Efforts will be made to understand the evolution of fractality in the structure of harmonic measure on the boundary of diffusion limited aggregates. Harmonic measure lies at the intersection of several important areas of analysis and probability. It seeks to measure fine structure of sets. It provides quantitative information of highly complex object like fractals and badly disconnected sets with hidden structures arising in the study of two-dimensional dynamics. ***

小行星9402946 本计画着重于数学分析中一个基本概念-调和测度的数学研究。 有几个等价的定义，从布朗路径到达固定集合的概率到势能理论中所理解的该集合的平衡分布。 近年来，调和测度的整体度量性质和特征，如最小Borel支撑的维数，绿色函数边界行为的例外集的大小，共形映射下维数畸变的界，与此同时，与解析动力系统的遍历理论也建立了密切的联系。 调和测度对于类似于Cantor集或Julia集的边界具有自然的动力学性质，遍历理论为研究调和测度的度量性质提供了有力的工具。 在相反的方向，调和测度的估计描述了分形的几何形状和动力学的某些性质。 本项目的主要目标是探索一般平面域以及一些重要的特殊分形类的调和测度的精细结构。 刻画调和测度整体度量性质的多重分形谱研究的继续。 应用朱莉娅集的各态历经理论处理相应压力函数的解析性问题和相变现象。 我们将致力于理解分形在有限扩散集合边界上的调和测度结构中的演化。 调和测度位于分析和概率的几个重要领域的交叉点。 它试图测量集合的精细结构。 它提供了在二维动力学研究中出现的具有隐藏结构的高度复杂对象（如分形和严重不连通集）的定量信息。 ***