Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published July 2017 | Submitted
Journal Article Open

Generic Stationary Measures and Actions

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.

Additional Information

© 2017 American Mathematical Society. Received by the editors February 17, 2015 and, in revised form, July 2, 2015, August 10, 2015, and August 14, 2015. Published electronically: January 9, 2017. 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.

Attached Files

Submitted - 1405.2260.pdf

Files

1405.2260.pdf
Files (472.2 kB)
Name Size Download all
md5:5be7efc1e0813a3d4cfb42c962d8e6c3
472.2 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
October 23, 2023