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