Mathematical Sciences: Multifractal Analysis of Harmonic Measure

在多种形式中，调和测度是复变函数理论、势论以及最近的解析动力学中出现的许多深层问题的核心。 定义特别简单。 对于任何维度的欧几里得空间中的给定域，函数被定义为在边界的给定子集上等于 1，在边界的其余部分上等于 0。 调和测度是具有这些边界值的域上的调和函数（或函数在任何指定点的值）。这也是从固定内点发出的布朗路径首先到达给定集合中的边界的概率。 本研究的主题是探索分形边界上调和测度的精细结构，并通过分形近似将其与一般域调和测度的某些随机性质联系起来。 这项工作开始了对所谓的维数谱的系统研究，它描述了谐波测量的特性。 它与随机过程中的大偏差理论有关。 主要研究课题是： 1. 维数谱的共性界限的估计。 2.分形近似理论。 3. 特定 Julia 集和其他重要分形类别的维数谱的属性和数值参数。

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.