Extremal holomorphic maps and the symmetrised bidisc

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\mathrm{\Gamma }$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence ${\mathcal{C}}_{\nu },\nu \ge 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that ${\mathcal{C}}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\ge 3$. We propose the conjecture that condition ${\mathcal{C}}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\mathrm{\Gamma }$ and we explore the implications of this conjecture. We introduce a classification of rational $\mathrm{\Gamma }$-inner functions, that is, analytic functions from the disc into $\mathrm{\Gamma }$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\mathrm{\Gamma }$. The classes are related to $n$-extremality and the conditions ${\mathcal{C}}_{\nu }$; we prove numerous strict inclusions between the classes.