Bull. Austral. Math. Soc. 72(3) pp.441--454, 2005.

The Hutchinson-Barnsley theory
for infinite iterated function systems

Gertruda Gwóźdź-Łukawska

Jacek Jachymski

Received: 19th July, 2005

We are grateful to Andrzej Komisarski for some useful discussion.


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 $ \in $ $ \mathbb {N}$} is a family of Matkowski's contractions on a complete metric space (X, d ) such that (Fi x0)i $\scriptstyle \in $ $\scriptstyle \mathbb {N}$ is bounded for some x0 $ \in $ X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, K = $ \bigcup \limits _{{i\in {\mathbb N}}}^{}$Fi (K). Moreover, given $ \si $ $ \in $ $ \mathbb {N}$$\scriptstyle \mathbb {N}$ and x $ \in $ X, the limit $ \lim \limits _{{n\ra \iy }}^{}$F$\scriptstyle \si _{1}$o ...oF$\scriptstyle \si _{n}$(x) 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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(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199645 Z'blatt-MATH: 1098.39015


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