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.
