Mathematical Sciences: Multifractal Analysis of Harmonic Measure

在它的许多伪装中，调和测度位于 复变函数论中出现的许多深层次问题， 理论，以及最近的分析动力学。 该定义 特别简单。 对于欧氏空间中的给定域， 在任何维度上，函数被定义为在给定的 边界的子集和边界的其余部分为零。 调和测度是定义域上调和函数 拥有这些边界值（或函数在 任何指定点）。这也是一个布朗运动 从一个固定的内部点发出的路径首先到达边界 在给定的集合中。 本研究的主旨是探讨 分形边界上调和测度结构及对 通过分形近似， 一般整环调和测度随机性质 这项工作开始了对所谓维度的系统研究 描述调和测度性质的谱。 是 与随机过程中的大偏差理论有关。 研究的主要课题是：1.泛界估计 关于维度光谱。 2.分形近似理论3. 维数谱的性质和数值参数 对于特定的Julia集和其他重要的 分形

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.