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The Hutchinson–Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

Volume 125 / 2023

Elismar R. Oliveira Banach Center Publications 125 (2023), 99-118 MSC: Primary 28A80; Secondary 54H20, 26E25, 47H04, 93E03. DOI: 10.4064/bc125-9


The study of generalized iterated function systems (GIFS) was introduced by Mihail and Miculescu in 2008. We provide a new approach to study those systems as the limit of the Hutchinson–Barnsley setting for infinite iterated function systems (IIFS) which has been developed by many authors in the last years. We show that any attractor of a contractive generalized iterated function system is the limit with respect to Hausdorff–Pompeiu metric of attractors of contractive infinite iterated function systems. We also prove that any Hutchinson measure for a contractive generalized iterated function system with probabilities is the limit with respect to the Monge–Kantorovich metric of the Hutchinson measures for contractive infinite iterated function systems with probabilities.


  • Elismar R. OliveiraInstitute of Mathematics and Statistics, Universidade Federal do Rio Grande do Sul
    Av. Bento Gonçalves, 9500, 43-111 Porto Alegre, Brazil

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