  We study finite state random dynamical systems (RDS) and their induced Markov
chains (MC) as stochastic models for complex dynamics. The linear
representation of deterministic maps in RDS are matrix-valued random variables
whose expectations correspond to the transition matrix of the MC. The
instantaneous Gibbs entropy, Shannon-Khinchin entropy per step, and the entropy
production rate of the MC are discussed. These three concepts as key anchor
points in stochastic dynamics, characterize respectively the uncertainties of
the system at instant time $t$, the randomness generated per step, and the
dynamical asymmetry with respect to time reversal. The entropy production rate,
expressed in terms of the cycle distributions, has found an expression in terms
of the probability of the deterministic maps with the single attractor in the
maximum entropy RDS. For finite RDS with invertible transformations, the
non-negative entropy production rate of its MC is bounded above by the
Kullback-Leibler divergence of the probability of the deterministic maps with
respect to its time-reversal dual probability.
