Relative stationary dynamical systems

TL;DR

This paper extends the concept of stationary dynamical systems to factor maps between systems, providing a structure theorem and establishing the existence and uniqueness of a relative Poisson boundary.

Contribution

It introduces a generalized notion of stationary factors for group actions and proves a structure theorem along with the existence and uniqueness of a relative Poisson boundary.

Findings

Generalized stationary factor maps for group actions

Proved a structure theorem extending Furstenberg-Glasner

Established existence and uniqueness of the relative Poisson boundary

Abstract

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.