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The Hutchinson-Barnsley theory for infinite iterated function systems

Published online by Cambridge University Press:  17 April 2009

Gertruda Gwóźdź-Lukawska  and
Jacek Jachymski
Gertruda Gwóźdź-Lukawska
Affiliation:
Centre of Mathematics and Physics, Technical University of Lódź, al. Politechniki 11, 90–924 Lódź, Poland, e-mail: gertruda@p.lodz.pl
Jacek Jachymski
Affiliation:
Institute of Mathematics Technical University of Lódź, Wólczańska 215, 93–005 Lódź, Poland, e-mail: jachym@p.lodz.pl
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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 x0 ∈ X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, . Moreover, given σ ∈ ℕℕ and x ∈ X, 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 441 - 454
Copyright
Copyright © Australian Mathematical Society 2005

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