Skip to main content
×
Home
    • Aa
    • Aa

The Hutchinson-Barnsley theory for infinite iterated function systems

  • Gertruda Gwóźdź-Lukawska (a1) and Jacek Jachymski (a2)
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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The Hutchinson-Barnsley theory for infinite iterated function systems
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      The Hutchinson-Barnsley theory for infinite iterated function systems
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      The Hutchinson-Barnsley theory for infinite iterated function systems
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax