Relative stationary dynamical systems

Abstract

Let $G$ be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure $\hat{I}¼$. We generalize the notion of $\hat{I}¼$-stationary systems to $\hat{I}¼$-stationary $G$-factor maps $\ddot{I}€:(X,\hat{I}½)\to (Y,\hat{I}\cdot )$. 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.