Generic stationary measures and actions

Let G be a countably infinite group, and let µ be a generating probability measure on G. We study the space of µ-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 µ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When µ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G, µ). When Z is compact, this implies that the simplex of µ-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 µ. 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.