How long is the chaos game?

Ian D. Morris; Natalia Jurga

doi:10.1112/blms.12539

How long is the chaos game?

Ian D. Morris^*, Natalia Jurga^*

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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Abstract

In the 1988 textbook Fractals Everywhere, Barnsley introduced an algorithm for generating fractals through a random procedure which he called the chaos game. Using ideas from the classical theory of covering times of Markov chains, we prove an asymptotic formula for the expected time taken by this procedure to generate a 𝛿 -dense subset of a given self-similar fractal satisfying the open set condition.

Original language	English
Pages (from-to)	1749-1765
Number of pages	17
Journal	Bulletin of the London Mathematical Society
Volume	53
Issue number	6
Early online date	20 Sept 2021
DOIs	https://doi.org/10.1112/blms.12539
Publication status	Published - Dec 2021

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10.1112/blms.12539Licence: CC BY

Morris_2021_BLMS_How-long-chaos-game_CC
Copyright © 2021 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Final published version, 417 KBLicence: CC BY

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

@article{9530738485a24a55af06b03a47a6b167,

title = "How long is the chaos game?",

abstract = "In the 1988 textbook Fractals Everywhere, Barnsley introduced an algorithm for generating fractals through a random procedure which he called the chaos game. Using ideas from the classical theory of covering times of Markov chains, we prove an asymptotic formula for the expected time taken by this procedure to generate a 훿 -dense subset of a given self-similar fractal satisfying the open set condition.",

author = "Morris, {Ian D.} and Natalia Jurga",

note = "Funding: Leverhulme Trust (Grant Number(s): RPG-2016-194), Engineering and Physical Sciences Research Council (Grant Number(s): EP/R015104/1).",

year = "2021",

month = dec,

doi = "10.1112/blms.12539",

language = "English",

volume = "53",

pages = "1749--1765",

journal = "Bulletin of the London Mathematical Society",

issn = "0024-6093",