Bulletin of the Australian Mathematical Society

Research Article

The Hutchinson-Barnsley theory for infinite iterated function systems

Gertruda Gwóźdź-Lukawskaa1 and Jacek Jachymskia2

a1 Centre of Mathematics and Physics, Technical University of Lódź, al. Politechniki 11, 90–924 Lódź, Poland, e-mail: gertruda@p.lodz.pl

a2 Institute of Mathematics Technical University of Lódź, Wólczańska 215, 93–005 Lódź, Poland, e-mail: jachym@p.lodz.pl

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 xs2208 xs2115} is a family of Matkowski's contractions on a complete metric space (X, d) such that (Fix0)ixs2208N is bounded for some x0 xs2208 X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, S0004972700035267_inline1. Moreover, given σ xs2208 xs2115xs2115 and x xs2208 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.

(Received July 19 2005)