3-extremal holomorphic maps and the symmetrised bidisc

TL;DR

This paper investigates 3-extremal holomorphic maps from the unit disc to the symmetrised bidisc, classifying them into two types, and provides methods to construct and analyze these maps using classical interpolation techniques.

Contribution

It introduces a classification of rational G-inner functions into aligned and caddywhompus types, and develops a construction method linking 3-extremal maps to Nevanlinna-Pick problems.

Findings

Identifies two classes of rational G-inner functions: aligned and caddywhompus.

Shows aligned functions are 3-extremal.

Reduces 3-point interpolation to classical Nevanlinna-Pick problems.

Abstract

We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be called {\em aligned} and {\em caddywhompus} functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational $\mathcal{G}$-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from $\mathbb{D}$ to $\mathcal{G}$ to a collection of classical Nevanlinna-Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for $\mathcal{G}$.