Mathematical Sciences: Harmonic measure and fractal sets

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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. ***