Random Backward Iteration Algorithm for Julia sets of Rational Semigroups

TL;DR

This paper extends a random backward iteration algorithm for visualizing Julia sets from single rational maps to finitely generated rational semigroups, broadening the scope of computational fractal generation.

Contribution

It proves that the algorithm for drawing Julia sets applies to finitely generated rational semigroups, not just individual rational maps, and discusses related implications.

Findings

Algorithm successfully visualizes Julia sets of rational semigroups.

Extension from single maps to semigroups verified.

Provides theoretical foundation for computational methods in complex dynamics.

Abstract

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely generated rational semigroups (of a certain type). We also provide some consequences of this result.