Contractive Markov systems II

Discrete time random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniform continuous and contractive are considered. A notion of a generating communication class of such a system is introduced, which includes every communication class if the system has a finite Markov partition. It is shown that the ergodic decomposition of an equilibrium state associated with such a system is purely atomic and can be exhaustively described using the generating communication classes if the system satisfies an absolute continuity condition (ACC). In such a case, each invariant Borel probability measure which is an image of an ergodic component of an equilibrium state under the coding map can be obtained by a random walk starting at any point in the corresponding generating communication class. As a by-product, a practical method for a computation of the entropy of the equilibrium states is obtained. Finally, it is shown that such a non-degenerate system satisfying the ACC which in addition has a dominating Markov chain and a finite (20) has a unique invariant Borel probability measure if and only if it has a single generating communication class. Some sufficient conditions for the ACC are provided.