On the convergence rate of the chaos game

TL;DR

This paper analyzes the convergence rate of the chaos game in relation to the Minkowski dimension of measures, providing bounds on cover times and characterizing probability vectors for Bedford-McMullen carpets.

Contribution

It establishes a link between the chaos game's convergence rate and Minkowski dimension, offering bounds and a novel characterization of probability vectors in Bedford-McMullen carpets.

Findings

Growth rate of cover time depends on Minkowski dimension.

Bounds on expected cover time with logarithmic corrections.

Characterization of probability vectors minimizing Minkowski dimension.

Abstract

This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen carpets we completely characterise the family of probability vectors which minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford-McMullen carpets before.