Algebraic and geometric aspects of rational \(\Gamma\)-inner functions (Q1705458)

The authors consider the open symmetrized bidisk \(\mathcal{G}:=\{(z+w,zw);z,w\in\mathbb{D}\}\subset\mathbb{C}^2\), where, as usually, \(\mathbb{D}\) is the unit disk in the complex plane \(\mathbb{C}\), whose closure is denoted by \(\Gamma\). This domain has special geometric properties, among which the authors signal the following: (1) the existence of a three-parameter group of automorphisms, (2) the boundary \(b\Gamma\) is homeomorphic to the Möbius band, and (3) a strict convex combination of \(\Gamma\)-inner functions is again a \(\Gamma\)-inner function. This framework is used by the authors to investigate the structure of the rational maps from the unit disk to \(\Gamma\), mapping the boundary of \(\mathbb{D}\) into the distinguished boundary of \(\Gamma\), obtaining properties which have no analog in the classical theory of inner functions.