The range and valence of a real Smirnov function

TL;DR

This paper characterizes the possible ranges and valences of real Smirnov functions, using advanced operator theory, Hardy spaces, and novel geometric constructions like disk and valence trees.

Contribution

It provides a complete description of the ranges and valences of real Smirnov functions, introducing disk and valence trees for their characterization.

Findings

Complete range characterization of real Smirnov functions

Finite valence characterization using disk and valence trees

Application of unbounded symmetric Toeplitz operator theory

Abstract

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.