Clark measures for rational inner functions II: general bidegrees and higher dimensions

TL;DR

This paper extends the theory of Clark measures to general rational inner functions in multiple variables, providing detailed descriptions of their support, weights, and conditions for unitarity of associated embeddings.

Contribution

It offers new characterizations of Clark measures for multi-variable rational inner functions, including those with singularities, and analyzes the unitarity of Clark embeddings in two variables.

Findings

Support sets and weights described via level sets and derivatives

Conditions for unitarity of Clark embeddings established

Relation between weights vanishing and contact order at singularities

Abstract

We study Clark measures associated with general two-variable rational inner functions (RIFs) on the bidisk, including those with singularities, and with general $d$-variable rational inner functions with no singularities. We give precise descriptions of support sets and weights for such Clark measures in terms of level sets and partial derivatives of the associated RIF. In two variables, we characterize when the associated Clark embeddings are unitary, and for generic parameter values, we relate vanishing of two-variable weights with the contact order of the associated RIF at a singularity.