I will discuss the various kinds of singularities that can occur on the boundary of self-maps of the polydisk in the context of so-called “Julia-Caratheodory theorems.” The classical Julia-Caratheodory establishes that any function from the disk to the disk which satisfies some mild asymptotic condition near a boundary point nontangentially actually has a nontangential limit, and, furthermore, has a derivative which is valid for directions pointing into the disk. Agler, McCarthy, and Young extended the Julia-Caratheodory theorem to the bidisk but noted nice behavior came in at least two strengths, which they called B-points and C-points. I will describe how non-tangential regularity is reflected in the so-called "realization theory" for a related class of functions from the bi-upper half plane into the upper half plane in terms of the "Hankel vector moment sequence" theory of Agler and McCarthy. |