Stability of the Markov operator and synchronization of Markovian random products

Abstract

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.

Attracting graphs of skew products with non-contracting fiber maps

“…random products. In [4] we prove that this property implies P(S F ) = 1 and hence, by Theorem 3, there is a global attracting measure v P ∈ M P . Theorem 4 states that rate of convergence to the attracting measure is exponential (with respect to the Wasserstein metric).…”

“…If, in addition, we assume that S F has probability one (note that this does not involve any contracting-like property) we obtain also ergodic information about the invariant graph, which is attracting from the ergodic point of view. In this context, invoking a variation of the "splitting property" of Dubins and Freedman [5] introduced in [4], we put Stark's result [18] into a much more general framework. In some sense, this topologicallike "splitting property" replaces in a topological way Stark's hypotheses on the Lyapunov exponent.…”

Random products of maps synchronizing on average

“…The key hypothesis in [17] is that the map in (1.3) is well-defined for P-almost every ω in Ω and it is proved that the distribution of Φ is the unique stationary measure. We point out that the property Φ is well-defined (almost everywhere) is, in fact, a consequence of much weaker topological conditions such as the so-called splitting property in [9] or the very classical condition of being contracting on average in [6]. For further results on the existence of invariant maps see the comments which follow Theorem 1.…”

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

“…Hence we get the splitting property f n 1 • f 2 ([0, 1]) ∩ f 2 ([0, 1]) = ∅ for n sufficiently large. Now Theorem 4.1 in [11], implies that S wh = ∅. Example 4.2 (An IFS without strict attractors and such that A tar is stable).…”