On the chaos game of Iterated Function Systems

TL;DR

This paper investigates the conditions under which attractors of iterated function systems can be generated by the chaos game, highlighting the importance of minimality conditions for deterministic rendering.

Contribution

It establishes necessary and sufficient conditions for the deterministic chaos game to generate attractors, especially on the circle, and explores the role of minimality and fibred quasi-attractors.

Findings

Quasi-attractors are renderable by the probabilistic chaos game.

Backward minimality is necessary for the deterministic chaos game.

Certain attractors on the circle are not deterministically renderable without both minimalities.

Abstract

Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.