On the convergence rate of the chaos game

Balázs Bárány*, Natalia Jurga, István Kolossváry

*Corresponding author for this work

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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.
Original languageEnglish
Article numberrnab370
Pages (from-to)4456-4500
Number of pages45
JournalInternational Mathematics Research Notices
Volume2023
Issue number5
Early online date25 Jan 2022
DOIs
Publication statusPublished - 1 Mar 2023

Keywords

  • Fractals
  • Minkowski dimension of measures
  • Chaos game
  • Self-affine carpets

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