Generic Stationary Measures and Actions

TL;DR

This paper investigates the structure and generic properties of stationary measures and actions for countably infinite groups, revealing conditions under which certain measures and actions are typical or generic in various topologies.

Contribution

It establishes that for groups with finite entropy measures, generic stationary measures are free extensions of the Poisson boundary, and characterizes the genericity of ergodic actions under different topologies.

Findings

Generic stationary measures are free extensions of the Poisson boundary when entropy is finite.

The simplex of stationary measures on $Z^G$ is a Poulsen simplex for compact $Z$.

Ergodic actions are meager for groups with property (T), but can be non-dense for some groups without property (T).

Abstract

Let $G$ be a countably infinite group, and let $\mu$ be a generating probability measure on $G$. We study the space of $\mu$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any perfect Polish space. We also study the space of $\mu$-stationary, measurable $G$-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When $\mu$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,\mu)$. When $Z$ is compact, this implies that the simplex of $\mu$-stationary measures on $Z^G$ is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on $\{0,1\}^G$. We furthermore show that if the action of $G$ on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when $G$ has property (T), the ergodic actions are meager. We also construct a group $G$ without property (T) such that the ergodic actions are not dense, for some $\mu$. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.