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 $\mathcal{ G}$.
Categories: math.CV