Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T00:01:34.701Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW EXAMPLE OF A DETERMINISTIC CHAOS GAME

Part of: Ergodic theory Real functions Classical measure theory

Published online by Cambridge University Press:  02 June 2016

ALIASGHAR SARIZADEH
ALIASGHAR SARIZADEH*
Affiliation:
Department of Mathematics, Ilam University, Ilam, Iran email [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a new necessary and sufficient condition for an iterated function system to satisfy the deterministic chaos game. As a consequence, we give a new example of an iterated function system which satisfies the deterministic chaos game.

Keywords

iterated function systemstrict attractordeterministic chaos gameforward and backward minimal

MSC classification

Primary: 26A18: Iteration
Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 37A99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 3 , December 2016 , pp. 464 - 470
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Barnsley, M. F., Fractals Everywhere (Academic Press, Orlando, FL, 1988).Google Scholar
Barnsley, M. F. and Leśniak, K., ‘The chaos game on a general iterated function system from a topological point of view’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24(11) (2014), 10 pages, 1450139.CrossRefGoogle Scholar
Barnsley, M. F., Leśniak, K. and Rypka, M., ‘Chaos game for IFSs on topological spaces’, J. Math. Anal. Appl. 435(2) (2016), 1458–1466.CrossRefGoogle Scholar
Barnsley, M. F. and Vince, A., ‘Developments in fractal geometry’, Bull. Aust. Math. Soc. 3 (2013), 299–348.Google Scholar
Barrientos, P. G., Fakhari, A., Malicet, D. and Sarizadeh, A., ‘Expanding actions: minimality and ergodicity’, arXiv:1307.6054 (2015).Google Scholar
Barrientos, P. G., Fakhari, A. and Sarizadeh, A., ‘Weakly hyperbolic and fibre-wise orbits’, in: 44th Annual Iranian Mathematics Conference (Ferdowski University of Mashhad, Iran, 2013), (extended abstract).Google Scholar
Barrientos, P. G., Fakhari, A. and Sarizadeh, A., ‘Density of fiberwise orbits in minimal iterated function systems on the circle’, Discrete Contin. Dyn. Syst. 34(10) (2014), 3341–3352.Google Scholar
Barrientos, P. G., Ghane, F. H., Malicet, D. and Sarizadeh, A., ‘On the chaos game of iterated function systems’, Topol. Methods Nonlinear Anal., to appear.Google Scholar
Barrientos, P. G., Ki, Y. and Raibekas, A., ‘Symbolic blender-horseshoes and applications’, Nonlinearity 27(12) (2014), 2805–2839.CrossRefGoogle Scholar
Barrientos, P. G. and Raibekas, A., ‘Dynamics of iterated function systems on the circle close to rotations’, Ergodic Theory Dynam. Systems 35(5) (2015), 1345–1368.Google Scholar
Calude, C. S., Priese, L. and Staiger, L., ‘Disjunctive sequences: an overview’, Research Report 63, University of Auckland, CDMTCS, Auckland, 1997.Google Scholar
Ghane, F. H., Homburg, A. J. and Sarizadeh, A., ‘ C 1 robustly minimal iterated function systems’, Stoch. Dyn. 10(1) (2010), 155–160.Google Scholar
Homburg, A. J. and Nassiri, M., ‘Robust minimality of iterated function systems with two generators’, Ergodic Theory Dynam. Systems 34(6) (2014), 1914–1929.Google Scholar
Leśniak, K., ‘Random iteration for infinite nonexpansive iterated function systems’, Chaos 25(8) (2015), 5 pages, 083117.Google Scholar
Nadler, S. B., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158 (Marcel Dekker, New York, 1992).Google Scholar
Palis, J. Jr and de Melo, W., Geometric Theory of Dynamical Systems: An Introduction (Springer, New York, 1982).Google Scholar
Staiger, L. and Sumi, H., ‘Random backward iteration algorithm for Julia sets of rational semigroups’, Discrete Contin. Dyn. Syst. 35 (2015), 2165–2175.Google Scholar