Nevanlinna domains with large boundaries

TL;DR

This paper demonstrates the existence of Nevanlinna domains with large, complex boundaries, including positive measure and arbitrary Hausdorff dimension, and constructs rational functions with extensive boundary lengths.

Contribution

It introduces new Nevanlinna domains with large boundaries and provides constructions of rational functions with extremely long boundaries for given poles.

Findings

Domains can have boundaries of positive planar measure.

Accessible points can have any Hausdorff dimension between 1 and 2.

Constructs rational functions with extremely long boundaries.

Abstract

We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between $1$ and $2$. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.