Mating, paper folding, and an endomorphism of PC^2

TL;DR

This paper explores the topological properties of Julia sets for a specific complex projective plane map and reveals a connection to uncountable families of paper folding curves that fill the plane.

Contribution

It introduces a novel link between complex dynamics and geometric paper folding curves, expanding understanding of Julia sets in projective spaces.

Findings

Julia set properties of the given map analyzed

Connection established between Julia sets and paper folding curves

Uncountable family of plane filling curves related to the dynamics

Abstract

We are studying topological properties of the Julia set of the map $F(z, p)=((2z/(p+1)-1)^2, ((p-1)/(p+1))^2)$ of the complex projective plane $PC^2$ to itself. We show a relation of this rational function with an uncountable family of "paper folding" plane filling curves.