Rational tetra-inner functions and the special variety of the tetrablock

TL;DR

This paper explores the complex geometry of the tetrablock, focusing on rational inner functions and their relation to a special subvariety called the royal variety, providing a detailed structure theory for these functions.

Contribution

It develops an explicit structure theory for rational -inner functions on the tetrablock, linking their behavior to the geometry of the royal variety and automorphisms.

Findings

Rational -inner functions either map into the royal variety or intersect it a number of times equal to their degree.

The paper characterizes the image of these functions and their intersections with the royal variety.

It analyzes convex subsets and extreme points within the set of all rational -inner functions.

Abstract

The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about $0$. It has a group of automorphisms parametrised by ${\mathrm{Aut}~} {\mathbb{D}} \times {\mathrm{Aut}~} {\mathbb{D}} \times {\mathbb{Z}}_2$ and its distinguished boundary $b\overline{\mathbb{E}}$ is homeomorphic to the solid torus $\overline{\mathbb{D}} \times {\mathbb{T}}$. It has a special subvariety \[\mathcal{R}_{\mathbb{\overline{E}}} = \big\{ (x_{1}, x_{2}, x_{3}) \in \overline{\mathbb{E}} : x_{1}x_{2}=x_{3} \big\}, \] called the royal variety of $\overline{\mathbb{E}}$, which is a complex geodesic of ${\mathbb{E}}$ that is invariant under all automorphisms of ${\mathbb{E}}$. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc ${\mathbb{D}}$ to $\overline{\mathbb{E}}$ that map the unit circle ${\mathbb{T}}$ to the distinguished boundary $b\overline{\mathbb{E}}$ of $\overline{\mathbb{E}}$. Such maps are called rational $\mathbb{ \overline{ E}}$-inner functions. We show that, for each nonconstant rational $\mathbb{ \overline{ E}}$-inner function $x$, either $x(\overline{\mathbb{D}}) \subseteq \mathcal{R}_{\mathbb{\overline{E}}} \cap \overline{\mathbb{E}}$ or $x(\overline{\mathbb{D}})$ meets $\mathcal{R}_{\mathbb{\overline{E}}}$ exactly $deg(x)$ times. We study convex subsets of the set $\mathcal{J}$ of all rational $\mathbb{ \overline{ E}}$-inner functions and extreme points of $\mathcal{J}$.