The Chaos Game on a General Iterated Function System

TL;DR

This paper proves that the chaos game algorithm can almost surely generate the attractor of a very broad class of iterated function systems, including non-contractive ones, under weaker conditions than previously known.

Contribution

It establishes a general theorem for the chaos game on iterated function systems, extending results to non-contractive systems with weaker assumptions.

Findings

The chaos game converges to the attractor under broad conditions.

The theorem applies to non-contractive iterated function systems.

Weaker conditions on the random orbit are sufficient for convergence.

Abstract

The main theorem of this paper establishes conditions under which the "chaos game" algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.