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Free minimal actions of countable groups with invariant probability measures

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  20 February 2020

GÁBOR ELEK
GÁBOR ELEK*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, LA1 4YF, UK email g.elek@lancaster.ac.uk
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Abstract

We prove that for any countable group $\unicode[STIX]{x1D6E4}$, there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

Keywords

free minimal actionsuniformly recurrent subgroupsinvariant measures

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1369 - 1389
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Abért, M. and Elek, G.. Hyperfinite actions on countable sets and probability measure spaces. Dynamical Systems and Group Actions (Contemporary Mathematics, 567) . American Mathematical Society, Providence, RI, 2012, pp. 1–16.Google Scholar
Bernshteyn, A.. Building large free subshifts using the Local Lemma. Groups Geom. Dyn. 13(4) (2019), 1417–1436.CrossRefGoogle Scholar
Elek, G.. Uniformly recurrent subgroups and simple C ∗ -algebras. J. Funct. Anal. 274(6) (2018), 1657–1689.Google Scholar
Epstein, I.. Orbit inequivalent actions of non-amenable groups. Preprint, 2007, arXiv:0707.4215.Google Scholar
Glasner, E. and Weiss, B.. Uniformly recurrent subgroups. Recent Trends in Ergodic Theory and Dynamical Systems (Contemporary Mathematics, 631) . American Mathematical Society, Providence, RI, 2015, pp. 63–75.Google Scholar
Hjorth, G. and Molberg, M.. Free continuous actions on zero-dimensional spaces. Topology Appl. 153(7) (2006), 1116–1131.CrossRefGoogle Scholar
Kechris, A. S., Solecki, S. and Todorcevic, S.. Borel chromatic numbers. Adv. Math. 141 (1999), 1–44.CrossRefGoogle Scholar
Matte Bon, N. and Tsankov, T.. Realizing uniformly recurrent subgroups. Ergod. Th. & Dynam. Sys. 40 (2020), 478–489.Google Scholar
Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.CrossRefGoogle Scholar
Seward, B. and Tucker-Drob, R. D.. Borel structurability on the 2-shift of a countable group. Ann. Pure Appl. Logic 167(1) (2016), 1–21.CrossRefGoogle Scholar
Weiss, B.. Minimal models for free actions. Dynamical Systems and Group Actions (Contemporary Mathematics, 567) . American Mathematical Society, Providence, RI, 2012, pp. 249–264.CrossRefGoogle Scholar