Algebraic and geometric aspects of rational $\Gamma$-inner functions

TL;DR

This paper explores the complex geometric properties of the set mma, focusing on rational maps from the unit disc to mma that preserve boundary conditions, revealing detailed structural insights.

Contribution

It develops an explicit structure theory for rational mma-inner functions, leveraging its geometric properties and automorphisms.

Findings

mma has a 3-parameter automorphism group.

Its distinguished boundary is a ruled surface homeomorphic to the M46bius band.

Identifies a unique complex geodesic invariant under all automorphisms.

Abstract

The set \[ \Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\"obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$.