Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-14T19:38:55.301Z Has data issue: false hasContentIssue false
  • English
  • Français

The chaos game on a general iterated function system

Published online by Cambridge University Press:  13 September 2010

MICHAEL F. BARNSLEY  and
ANDREW VINCE
MICHAEL F. BARNSLEY
Affiliation:
Department of Mathematics, Australian National University, Canberra, ACT, Australia (email: michael.barnsley@maths.anu.edu.au)
ANDREW VINCE
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA (email: avince@ufl.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1073 - 1079
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barnsley, M. F., Vince, A. and Wilson, D. C.. Real projective iterated function systems. Preprint, 2010.Google Scholar
[2]Barnsley, M. F.. Fractals Everywhere. Academic Press, Boston, MA, 1988.Google Scholar
[3]Barnsley, M. F. and Demko, S. G.. Iterated function systems and the global construction of fractals. Proc. R. Soc. Lond. Ser. A 399 (1985), 243–275.Google Scholar
[4]Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S.. Invariant measures arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré 24 (1988), 367–394.Google Scholar
[5]Berger, M. A.. Random affine iterated function systems: curve generation and wavelets. SIAM Rev. 34 (1992), 361–385.CrossRefGoogle Scholar
[6]McGehee, R. P. and Wiandt, T.. Conley decomposition for closed relations. Difference Equations Appl. 12 (2006), 1–47.CrossRefGoogle Scholar
[7]Elton, J. H.. An ergodic theorem for iterated maps. Ergod. Th. & Dynam. Sys. 7 (1987), 481–488.CrossRefGoogle Scholar
[8]Jaroszewska, J.. Iterated function systems with continuous place dependent probabilities. Univ. Iagel. Acta Math. 40 (2002), 137–146.Google Scholar
[9]Forte, B. and Mendivil, F.. A classical ergodic property for IFS: a simple proof. Ergod. Th. & Dynam. Sys. 18 (1998), 609–611.CrossRefGoogle Scholar
[10]Leśniak, K.. Stability and invariance of multivalued iterated function systems. Math. Slovaca 53 (2003), 393–405.Google Scholar
[11]Onicescu, O. and Mihoc, G.. Sur les chaǐnes de variables statistiques. Bull. Sci. Math. France 59 (1935), 174–192.Google Scholar
[12]Stenflo, Ö.. Uniqueness of invariant measures for place-dependent random iterations of functions. Fractals in Multimedia (IMA Volumes in Mathematics and its Applications, 132). Eds. Barnsley, M. F., Saupe, D. and Vrscay, E. R.. Springer, New York, 2002.Google Scholar
[13]Werner, I.. Ergodic theorem for contractive Markov systems. Nonlinearity 17 (2004), 2303–2313.Google Scholar
[14]Scealy, R.. V-variable fractals and interpolation. PhD Thesis, Australian National University, 2008.Google Scholar