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Generic Stationary Measures and Actions

Bowen, Lewis and Hartman, Yair and Tamuz, Omer (2017) Generic Stationary Measures and Actions. Transactions of the American Mathematical Society, 369 (7). pp. 4889-4929. ISSN 0002-9947. doi:10.1090/tran/6803.

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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.

Item Type:Article
Related URLs:
URLURL TypeDescription DOIArticle Paper
Tamuz, Omer0000-0002-0111-0418
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.
Funding AgencyGrant Number
Binational Science Foundation (USA-Israel)2008274
European Research Council (ERC)239885
Subject Keywords:stationary action, Poisson boundary
Issue or Number:7
Classification Code:2010 Mathematics Subject Classification: Primary 37A35
Record Number:CaltechAUTHORS:20161114-092928470
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71981
Deposited By: Ruth Sustaita
Deposited On:15 Nov 2016 23:55
Last Modified:11 Nov 2021 04:54

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