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Non-hyperbolic iterated function systems: semifractals and the chaos game

Volume 250 / 2020

Lorenzo J. Díaz, Edgar Matias Fundamenta Mathematicae 250 (2020), 21-39 MSC: 37C70, 28A80, 47H10. DOI: 10.4064/fm635-9-2019 Published online: 23 December 2019


We consider iterated function systems (IFSs) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a role similar to that played by semifractals introduced by Lasota and Myjak for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS to have a global attractor. We provide an example of a non-regular IFS whose target set is non-empty, showing that our approach gives rise to a new class of semifractals. Finally, we show that random orbits generated by IFSs draw target sets that are “stable”.


  • Lorenzo J. DíazDepartamento de Matemática
    Marquês de São Vicente 225, Gávea
    Rio de Janeiro 22451-900, Brazil
  • Edgar MatiasDepartamento de Matemática
    Universidade Federal da Bahia
    Av. Adhemar de Barros s/n
    40170-110 Salvador, Brazil

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