Intertwinings of tensor products, and the stochastic centralizer of dynamical systems

A dynamical system is called $\omega$-simple if all its ergodic joinings of the second order (except for $\mu \otimes \mu$) are measures concentrated on the graphs of finite-valued maps commuting with the system, the number of inequivalent graphs of this kind being at most countable. This class of dynamical systems contains, for example, horocycle flows and mixing actions of the group $\mathbb R^n$ with partial cyclic approximation. It is proved in this paper that $\omega$-simple mixing flows have multiple mixing, which is a consequence of results on stochastic intertwinings of flows. Properties of dynamical systems with general time are investigated in this direction, including actions with discrete and non-commutative time. The results obtained depend on the type of system.