On the Convergence Rate of the Chaos Game

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