A non-logarithmic approach to the rate of convergence of the deterministic chaos game

TL;DR

This paper offers a new perspective on the convergence rate of the chaos game algorithm, showing that typical drivers can yield arbitrarily slow or comparable rates, extending previous theoretical results.

Contribution

It introduces a non-logarithmic approach to analyze the convergence rate, broadening understanding of typical behaviors in the chaos game algorithm.

Findings

Typical drivers can yield arbitrarily slow convergence rates.

A typical driver can produce a rate of recovery comparable to any diverging function.

The approach extends previous theorems to a broader class of convergence behaviors.

Abstract

The aim of this paper is to provide a different perspective in the study of the rate of convergence of the chaos game algorithm to the attractor of an iterated function system. We prove that for any function $\psi$ with $\lim\limits_{\ve\to 0}\psi(\ve)=\infty$, a typical (in the sense of the Baire category) driver yields a rate of recovery comparable to $\psi$. This result extends the main theorem from Le\'sniak et al. (Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 157, 2024). Moreover, thanks to the change of perspective, we are able to prove that a typical driver gives arbitrarily slow rate of recovery.