On the Convergence Rate of the Chaos Game. (25th January 2022)

• Record Type: Journal Article
• Authors: Bárány, Balázs; Jurga, Natalia; Kolossváry, István
• Is Part Of: International mathematics research notices. Volume 2023:Number 5(2023)
• Page Start: 4456
• Page End: 4500
• Publication Date: 2022-01-25
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab370
• Languages: English
• ISSNs: 1073-7928

Abstract

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 IFS 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 that minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford–McMullen carpets before.