For a measure-preserving transformation, its direct product is naturally
defined and it is used to study ergodic-theoretic properties of the original
transformation. For example, a measure-preserving transformation
is weak-mixing if and only if its direct product is ergodic.
In the case of a random dynamical system given by random iteration of a family
of nonsingular transformations, we can consider the random iteration of
the direct products of transformations in the family.
I would like to talk about some ergodic-theoretic properties of such a random
iteration of direct products and their application to the study of sample-wise
ergodic behavior of the original random dynamical system.