Harmonic Measure Versus Hausdorff Measure on Uniformly Perfect Sets; Geometric Properties of Julia Sets

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

ABSTRACT: The goal of the project is to establish the singularity of harmonic measure and of Hausdorff measure on some totally disconnected, uniformly perfect sets (subsets of the real line or Julia sets of rational functions). The results rely on estimates of logarithmic capacities, Green's function and harmonic functions near the boundary of plane domains. It combines potential theoretic arguments with ergodic theory and dynamical systems,and uses techniques introduced by Carleson, Jones and Wolff, Makarov and Bourgain. The fractal sets I will study represent the chaotic part of a dynamical system. Almost all the phenomenon in life are dynamical systems: complex ones like weather, earthquakes and spreading of disease, and simple ones, like the game of billiard. Harmonic functions and potential theory were first of interest to physicists and engineers, particularly in the fields of thermodynamics and electro-magnetics. All the successes in waves today stem from this theory. The transmission of signals electro-magnetically between antennas, the design of antennas and their placement in arrays is one of the current problems that apply potential theory.