  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.
