Mathematical Sciences: Multifractal Analysis of Harmonic Measure

在它的许多形式中，调和测度是复数函数理论、势理论和最近的分析动力学中出现的许多深层问题的核心。这个定义特别简单。对于任意维欧几里得空间中的给定定义域，定义一个函数在边界的给定子集上等于1，在边界的其余部分上等于0。调和测度是在具有这些边值的域上的调和函数（或函数在任意指定点的值）。它也是一条布朗路径从一个固定的内部点发出首先到达给定集合边界的概率。本研究的主要主题是探讨分形边界上调和测度的精细结构，并利用分形逼近的方法将其与一般域调和测度的某些随机性质联系起来。这项工作开始了对描述谐波测度性质的所谓维谱的系统研究。这与随机过程中的大偏差理论有关。主要的研究课题有：1.；维谱的共界估计。2. 分形近似理论。3. 特殊Julia集和其他重要分形类的维数谱的性质和数值参数。

In its many guises, harmonic measure lies at the heart of many deep problems arising in complex function theory, potential theory and, of late, analytic dynamics. The definition is particularly simple. For a given domain in Euclidean space of any dimension, a function is defined to equal one on a given subset of the boundary and zero on the rest of the boundary. Harmonic measure is that harmonic function on the domain possessing these boundary values (or the value of the function at any specified point). It is also the probability that a Brownian path issuing from a fixed interior point first hits the boundary in the given set. The main theme of this research is to explore the fine structure of harmonic measure on fractal boundaries and to relate it, by means of fractal approximation, to certain stochastic properties of harmonic measure of general domains. This work begins a systematic study of the so-called dimension spectrum which describes properties of harmonic measure. It is relevant to the large deviation theory in stochastic processes. The main topics of study are: 1. Estimates of universals bounds for the dimension spectrum. 2. Fractal approximation theory. 3. Properties and numerical parameters of the dimension spectrum for particular Julia sets and other important classes of fractals.