Continuity and growth of free multiplicative convolution semigroups

Abstract

Let $μ$ be a compactly supported probability measure on the positive half-line and let $μ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $μ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.

11 pages