On the convergence rate of the chaos game
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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.
Bárány , B , Jurga , N & Kolossváry , I 2023 , ' On the convergence rate of the chaos game ' , International Mathematics Research Notices , vol. 2023 , no. 5 , rnab370 , pp. 4456-4500 . https://doi.org/10.1093/imrn/rnab370
International Mathematics Research Notices
Copyright © 2022 the Author(s). Published by Oxford University Press. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1093/imrn/rnab370
DescriptionFunding: Balázs Bárány acknowledges support from grants OTKA K123782 and OTKA FK134251. 759/1). Natalia Jurga was supported by an EPSRC Standard Grant (EP/R015104/1). István Kolossváry was supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).
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