Some stochastic Property of topological action semigroups

Under suitable conditions, with respect to some property on a random iterated function system(RIFS), it is shown how the system satisfies in this property almost surely.Comment: 9 page

On the chaos game of iterated function systems

Every quasi-attractor of an iterated function system (\rom{IFS}) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an \rom{IFS} of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal \rom{IFS} on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive \rom{IFS}s