On the supports of measures in free multiplicative convolution semigroups

Abstract

In this paper, we study the supports of measures in free multiplicative semigroups on the positive real line and on the unit circle. We provide formulas for the densities of the absolutely continuous parts of measures in these semigroups. The descriptions of the densities rely on the characterizations of the images of the upper half-plane and the unit disc under certain subordination functions. These subordination functions are \(\eta \)-transforms of infinitely divisible measures with respect to free multiplicative convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter.