On regularity for measures in multiplicative free convolution semigroups

TL;DR

This paper extends regularity results for measures in free convolution semigroups from additive to multiplicative cases, showing that certain intervals must contain mass, using subordination techniques.

Contribution

It generalizes previous additive free convolution regularity results to multiplicative free convolution semigroups, providing new insights into their measure properties.

Findings

Semigroups related to multiplicative free convolution cannot have gaps with no mass between atoms.

The proof employs subordination results to establish measure regularity.

Extension of known additive results to the multiplicative setting.

Abstract

Given a probability measure $\mu$ on the real line, there exists a semigroup $\mu_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $\mu_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that $\mu_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.