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On the support of measures in multiplicative free convolution semigroups
In this paper, we study the supports of measures in multiplicative free
semigroups on the positive real line and on the unit circle. We provide
formulas for the density of the absolutely continuous parts of measures in
these semigroups. The descriptions rely on the characterizations of the images
of the upper half-plane and the unit disc under certain subordination
functions. These subordination functions are -transforms of infinitely
divisible measures with respect to multiplicative free 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
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