Invariant measures and entropies of random dynamical systems and the variational principle for random Bernoulli shifts (Q804040)

Consider a product of random continuous maps of a compact metric space, where the randomness enters in a stationary and ergodic way. This set-up is called a (discrete time) random dynamical system (RDS). An RDS can be thought of as an ordinary dynamical system with a factor, where an invariant measure of the factor is given a priori. In the present paper, first a detailed proof for existence of invariant measures for RDS is presented and the ergodic decomposition of ``Markov measures'' is characterized. Topological and metric entropy of an RDS are introduced (essentially these are `fiber entropies'), and the Shannon-McMillan- Breiman theorem for the metric entropy of an RDS is established. Assuming compactness of the probability space, the variational principle for metric and topological entropy of RDS is deduced from Ledrappier and Walters' result [\textit{F. Ledrappier} and \textit{P. Walters}, J. Lond. Math. Soc., II. Ser. 16, 568-576 (1977; Zbl 0388.28020)]. A particular class of `random Bernoulli shifts' is introduced, for which uniqueness of the maximal measure is established. The general variational principle for RDS without topological assumptions on the probability space has been proved by T. Bogenschütz (Bremen) in an as yet unpublished paper.