Partially defined semigroups relative to multiplicative free convolution

Nica and Speicher have shown that every probability measure on the line belongs to a partial semigroup {μt : t ≥ 1} relative to additive free convolution (i.e., μt+s=μt⊞μs for t, s≥1). We prove analogous results for multiplicative free convolution on the positive half-line and on the circle. The existence of semigroups is derived from certain global inversion results for analytic functions defined on the disk or on the slit complex plane. A close analysis of the global inverses yields regularity results for the measures in these semigroups. We also use this analysis to improve the known results for the additive situation.