Abstract
We explore when the chaos game renders a quasi attractor of an iterated function system (IFS); we emphasize the role of the set in which the chaos game is initialized. For IFS with lower semicontinuous Hutchinson-Barnsley operator we find that these initial points necessarily belong to the largest invariant set of the weak basin of the attractor. Under additional assumptions, namely the attractor is a stable small attractor and the space compact, we find that the probabilistic chaos game with initial points in the largest invariant set of the weak basin does in fact render the attractor. We note that this result applies to IFS which contain discontinuous functions. Under the same additional assumptions we provide the same conclusion for the disjunctive chaos game played with an evenly continuous IFS (this is the topological version of equicontinuous IFS).
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