Generic stationary measures and actions
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- by Lewis Bowen, Yair Hartman and Omer Tamuz PDF
- Trans. Amer. Math. Soc. 369 (2017), 4889-4929 Request permission
Abstract:
Let $G$ be a countably infinite group, and let $\mu$ be a generating probability measure on $G$. We study the space of $\mu$-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 $\mu$-stationary, measurable $G$-actions on a standard, nonatomic probability space.
Equip the space of stationary measures with the weak* topology. When $\mu$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,\mu )$. When $Z$ is compact, this implies that the simplex of $\mu$-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 $\mu$.
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.
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Additional Information
- Lewis Bowen
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 671629
- Yair Hartman
- Affiliation: Weizmann Institute of Science, Rehovot, Israel
- Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60201
- MR Author ID: 1001252
- Omer Tamuz
- Affiliation: California Institute of Technology, Pasadena, California 91125
- MR Author ID: 898902
- Received by editor(s): February 17, 2015
- Received by editor(s) in revised form: July 2, 2015, August 10, 2015, and August 14, 2015
- Published electronically: January 9, 2017
- Additional Notes: 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. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4889-4929
- MSC (2010): Primary 37A35
- DOI: https://doi.org/10.1090/tran/6803
- MathSciNet review: 3632554