Computer Generated Images for Quadratic Rational Maps with a Periodic Critical Point

TL;DR

This paper presents an algorithm to identify hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point, visualizes these components, and analyzes the algebraic properties of specific parameter spaces.

Contribution

The authors develop a novel algorithm for distinguishing hyperbolic components and provide visualizations, along with algebraic analysis of parameter spaces V1-V5.

Findings

Generated computer images of hyperbolic components for V1-V4.

Resolved singularities of V5, showing its genus is 1.

Explained the difficulty in visualizing V5 due to its algebraic properties.

Abstract

We describe an algorithm for distinguishing hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point. We then illustrate computer images of the hyperbolic components of the parameter spaces V1 - V4, which were produced using our algorithm. We also resolve the singularities of the projective closure of V5 by blowups, giving an alternative proof that as an algebraic curve, the geometric genus of V5 is 1. This explains why we are unable to produce an image for V5.