On the supports of measures in free additive and multiplicative semigroups.

摘要

Given a Borel probability measure on the real line, consider its associated partially defined one-parameter free additive convolution semigroup. In this thesis, we study the supports of measures in the semigroup. We provide formulas for the densities of the absolutely continuous parts, with respect to the Lebesgue measure, of these measures. The descriptions of the densities rely on the characterizations of the images of the upper half-plane under certain subordination functions. These subordination functions are certain type of transform of infinitely divisible measures with respect to free additive convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the numbers of components in the supports of these measures and their absolutely continuous parts are both non-increasing functions of the parameter. We also give necessary and sufficient conditions on the given measure so that the supports of the measures in the semigroup associated to the given measure have only one component for large parameter. A measure such that measures in its associated semigroup have infinitely many components in their supports is given as well. We also investigate free multiplicative semigroups associating to measures on the positive real and the circle, and show that similar statements also hold.