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  • Bulletin of the Australian Mathematical Society, Volume 72, Issue 3
  • December 2005, pp. 441-454

The Hutchinson-Barnsley theory for infinite iterated function systems

  • Gertruda Gwóźdź-Lukawska (a1) and Jacek Jachymski (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700035267
  • Published online: 01 April 2009
Abstract

We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {Fi : i ∈ ℕ} is a family of Matkowski's contractions on a complete metric space (X, d) such that (Fix0)i∈N is bounded for some x0X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, . Moreover, given σ ∈ ℕ and xX, the limit exists and does not depend on x. We also study separately the case in which (X, d) is Menger convex or compact. Finally, we answer a question posed by Máté concerning a finite iterated function system {F1,…, FN} with the property that each of Fi has a contractive fixed point.

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[1]J. Andres and J. Fišer , ‘Metric and topological multivalued fractals’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), 12771289.

[2]J. Andres , J. Fišer , G. Gabor and K. Leśniak , ‘Multivalued fractals,’ Chaos Solitons Fractals 24 (2005), 665700.

[3]J. Andres and L. Górniewicz , ‘On the Banach contraction principle for multivalued mappings’, in Approximation, optimization and mathematical economics (Pointe-à Pitre) (Physica, Heidelberg, 2001), pp. 123.

[9]A. Granas and J. Dugundji , Fixed point theory, Springer Monographs in Mathematics (Springer-Verlag, New York, 2003).

[11]J. E. Hutchinson , ‘Fractals and self-similarity’, Indiana Univ. Math. J. 30 (1981), 713747.

[12]J. Jachymski , ‘Equivalence of some contractivity properties over metrical structures’, Proc. Amer. Math. Soc. 125 (1997), 23272335.

[13]J. Jachymski , ‘An extension of A. Ostrovski's theorem on the round-off stability of iterations’, Aequationes Math. 53 (1997), 242253.

[15]L. Máté , ‘The Hutchinson-Barnsley theory for certain non-contraction mappings’, Period. Math. Hungar. 27 (1993), 2133.

[18]J. Matkowski , ‘Fixed point theorem for mappings with a contractive iterate at a point’, Proc. Amer. Math. Soc. 62 (1977), 344348.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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