On the chaos game of Iterated Function Systems

Every attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space satisfies the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. We obtain that an IFS of homeomorphisms of the circle satisfies the deterministic chaos game if and only if it is forward and backward minimal. This provides examples of attractors that do not satisfy the deterministic chaos game. We also prove that every contractible attractor (in particular strong-fibred attractors) satisfies the deterministic chaos game.