Relative stationary dynamical systems

Let $G$ be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure $\mu$.

We generalize the notion of $\mu$-stationary systems to $\mu$-stationary $G$-factor maps $\pi: (X,\nu)\to (Y,\eta)$.

For these stationary relations between dynamical systems, we provide a structure theorem, which generalizes the structure theorem of Furstenberg-Glasner.

Furthermore, we show the existence and uniqueness of a relative version of the Poisson boundary in this setup.