The Hutchinson-Barnsley theory for infinite iterated function systems

E-ISSN： 1755-1633|72|3|441-454

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.72, Iss.3, 2005-12, pp. : 441-454

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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 {F_i : i } is a family of Matkowski's contractions on a complete metric space (X, d) such that (F_ix₀)iN 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 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,…, F_N} with the property that each of F_i has a contractive fixed point.