Color Visualization of Blaschke Self-Mappings of the Real Projective Plan

TL;DR

This paper explores the visualization of Blaschke self-mappings on the real projective plane, revealing their structure as non-orientable Klein surfaces through advanced continuation techniques.

Contribution

It introduces a method for visualizing Blaschke self-mappings on the real projective plane using simultaneous continuation, linking complex analysis with geometric visualization.

Findings

Visualization of Blaschke mappings on P^2 achieved

Identification of Klein surface structures in mappings

Application of continuation techniques to complex mappings

Abstract

The real projective plan $P^2$ can be endowed with a dianalytic structure making it into a non orientable Klein surface. Dianalytic self-mappings of that surface are projections of analytic self-mappings of the Riemann sphere $\widehat{\mathbb{C}}$. It is known that the only analytic bijective self-mappings of $\widehat{\mathbb{C}}$ are the Moebius transformations. The Blaschke products are obtained by multiplying particular Moebius transformations. They are no longer one-to-one mappings. However, some of these products can be projected on $P^2$ and they become dianalytic self-mappings of $P^2$. More exactly, they represent canonical projections of non orientable branched covering Klein surfaces over $P^2$. This article is devoted to color visualization of such mappings. The working tool is the technique of simultaneous continuation we introduced in previous papers.