## Abstract

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.

Original language | English |
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Pages (from-to) | 4889-4929 |

Number of pages | 41 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jan 2017 |

Externally published | Yes |

## Keywords

- Poisson boundary
- Stationary action

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics