We study Markovian random products on a large class of "mdimensional" connected compact metric spaces (including products of closed intervals and trees). We introduce a splitting condition, generalizing the classical one by Dubins and Freedman, and prove that this condition implies the asymptotic stability of the corresponding Markov operator and (exponentially fast) synchronization.2000 Mathematics Subject Classification. 37B25, 37B35, 60J05, 47B80.

We consider iterated function systems IFS(T 1 , . . . , T k ) consisting of continuous self maps of a compact metric space X. We introduce the subset St of weakly hyperbolic sequences ξ = ξ 0 . . . ξn . . .The target set π(St) plays a role similar to the semifractal introduced by Lasota-Myjak.Assuming that St = ∅ (the only hyperbolic-like condition we assume) we prove that the IFS has at most one strict attractor and we state a sufficient condition guaranteeing that the strict attractor is the closure of the target set. Our approach applies to a large class of genuinely non-hyperbolic IFSs (e.g. with maps with expanding fixed points) and provides a necessary and sufficient condition for the existence of a globally attracting fixed point of the Barnsley-Hutchinson operator. We provide sufficient conditions under which the disjunctive chaos game yields the target set (even when it is not a strict attractor).We state a sufficient condition for the asymptotic stability of the Markov operator of a recurrent IFS. For IFSs defined on [0, 1] we give a simple condition for their asymptotic stability. In the particular case of IFSs with probabilities satisfying a "locally injectivity" condition, we prove that if the target set has at least two elements then the Markov operator is asymptotically stable and its stationary measure is supported in the closure of the target set.

We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do 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 having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of semifractals. Finally, we show that random orbits generated by IFSs draws target sets that are "stable".2000 Mathematics Subject Classification. 37C70, 28A80, 47H10.

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.