Bulletin of the Australian Mathematical Society

Research Article

The Tarski–Kantorovitch prinicple and the theory of iterated function systems

Jacek Jachymskia1, Leslaw Gajeka1 and Piotr Pokarowskia2

a1 Institute of Mathematics, Technical University of Łódź, Al. Politechniki 11, 90-924 Łódź, Poland e-mail: jachm@ck-sg.p.lodz.pl gal@ck-sg.p.lodz.pl

a2 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland e-mail: pokar@hydra.mimuw.edu.pl

Abstract

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.

(Received May 31 1999)