Generic stationary measures and actions

Abstract

Let G G be a countably infinite group, and let μ \mu be a generating probability measure on G G . We study the space of μ \mu -stationary Borel probability measures on a topological G G space, and in particular on Z G Z^G , where Z Z is any perfect Polish space. We also study the space of μ \mu -stationary, measurable G G -actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When μ \mu has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of ( G , μ ) (G,\mu ) . When Z Z is compact, this implies that the simplex of μ \mu -stationary measures on Z G Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on { 0 , 1 } G \{0,1\}^G . We furthermore show that if the action of G 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 G has property (T), the ergodic actions are meager. We also construct a group G G without property (T) such that the ergodic actions are not dense, for some μ \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.