Relative stationary dynamical systems

Let G be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure μ. We generalize the notion of μ-stationary systems to μ-stationary G-factor maps π: (X,ν)→ (Y,η). 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.