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3-extremal holomorphic maps and the symmetrised bidisc

Lookup NU author(s): Professor Jim Agler, Dr Zinaida Lykova, Professor Nicholas Young

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).


Abstract

We analyse the $3$-extremal holomorphic maps from the unit disc $\D$ to the symmetrised bidisc $\G \df \{(z+w,zw): z,w\in\D\}$ with a view to the complex geometry and function theory of $\G$. These are the maps whose restriction to any triple of distinct points in $\D$ 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 $\G$-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 $\G$-inner functions; with the aid of this method we reduce the solution of a $3$-point interpolation problem for aligned holomorphic maps from $\D$ to $\G$ to a collection of classical Nevanlinna-Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for $\G$.


Publication metadata

Author(s): Agler J, Lykova ZA, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Geometric Analysis

Year: 2015

Volume: 25

Issue: 3

Pages: 2060-2102

Print publication date: 01/07/2015

Online publication date: 15/07/2014

ISSN (print): 1050-6926

ISSN (electronic): 1559-002X

Publisher: Springer

URL: http://dx.doi.org/10.1007/s12220-014-9504-3

DOI: 10.1007/s12220-014-9504-3


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