Algebraic and geometric aspects of rational Γ-inner functions

Jim Agler,Zinaida A Lykova,N.J Young

doi:10.1016/j.aim.2017.12.018

Jim Agler, Zinaida A Lykova + Show 1 more

Open Access

https://doi.org/10.1016/j.aim.2017.12.018

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Abstract

The setG=def{(z+w,zw):|z|<1,|w|<1}⊂C2 has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius band and it has a special subvariety which is the only complex geodesic of G that is invariant under all automorphisms. We exploit the geometry of G to develop an explicit and detailed structure theory for the rational maps from the unit disc to the closure Γ of G that map the boundary of the disc to the distinguished boundary of Γ.

Highlights

Recall that a rational inner function is a rational map h from the open unit disc D in the complex plane C to its closure D− with the property that h maps the unit circle T into itself
The appropriate analog of rational inner functions is likely to play a significant role in such a theory
We denote the closure of G by Γ and we define a rational Γ-inner function to be a rational analytic map h : D → Γ with the property that h maps T into the distinguished boundary bΓ of Γ

Summary

Introduction

Recall that a rational inner function is a rational map h from the open unit disc D in the complex plane C to its closure D− with the property that h maps the unit circle T into itself. We denote the closure of G by Γ and we define a rational Γ-inner function to be a rational analytic map h : D → Γ with the property that h maps T into the distinguished boundary bΓ of Γ. The second main result describes the construction of rational Γ-inner functions of prescribed degree from certain interpolation data; one can think of this result as an analog of the expression for a finite Blaschke product in terms of its zeros. A Γ-inner function of degree n having k royal nodes in T, counted with multiplicity, is an extreme point of the set of rational Γ-inner functions if and only if 2k > n. A Γ-inner function h is an extreme point of the set of rational Γ-inner functions if and only if the curve h(eit) touches the edge of the Möbius band bΓ more than.

Properties of Γ and Γ-inner functions

Royal nodes and the royal polynomial

Findings

Prescribing the royal nodes of h and zeros of s