TY - JOUR

T1 - Generic stationary measures and actions

AU - Bowen, Lewis

AU - Hartman, Yair

AU - Tamuz, Omer

N1 - Funding Information:
The first author was supported in part by NSF grant DMS-0968762, NSF CAREER Award DMS-0954606 and BSF grant 2008274. The second author was supported by the European Research Council, grant 239885. The authors are grateful to Itaï Ben Yaacov, Julien Melleray and Todor Tsankov for helping them understand the different topologies on the group Aut*(X, λ) of nonsingular transformations of a Lebesgue space. Part of this paper was written while all three authors attended the trimester program "Random Walks and Asymptotic Geometry of Groups" at the Henri Poincaré Institute in Paris. They are grateful to the Institute for its support.
Publisher Copyright:
© 2017 American Mathematical Society.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 ZG, 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 ZG 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.

AB - 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 ZG, 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 ZG 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.

KW - Poisson boundary

KW - Stationary action

UR - http://www.scopus.com/inward/record.url?scp=85017313816&partnerID=8YFLogxK

U2 - 10.1090/tran/6803

DO - 10.1090/tran/6803

M3 - Article

AN - SCOPUS:85017313816

SN - 0002-9947

VL - 369

SP - 4889

EP - 4929

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 7

ER -