Color Visualization of Blaschke Product Mappings

TL;DR

This paper presents an improved visualization technique for Blaschke product mappings using coloring of pre-images of annuli, enhancing the fidelity of the representations by refining the simultaneous continuation method.

Contribution

It refines the simultaneous continuation technique specifically for certain Blaschke products to produce more accurate and detailed visualizations.

Findings

Enhanced visualization accuracy for Blaschke products.

Effective coloring method for pre-images of annuli.

Improved understanding of covering Riemann surfaces.

Abstract

A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which coloring pre-images of annuli centered at the origin allow us to describe the mappings with a high degree of fidelity.