Measures and stabilizers of group Cantor actions

Maik Gröger; Olga Lukina

doi:10.3934/dcds.2020350

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2021, Volume 41, Issue 5: 2001-2029. Doi: 10.3934/dcds.2020350

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$ G^{\delta, 1} $

almost conservation law for mCH and the evolution of its radius of spatial analyticity

Measures and stabilizers of group Cantor actions

Maik Gröger^1,2, and
Olga Lukina^1, ,

1.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2.
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30-348 Kraków, Poland

^* Corresponding author
^* Corresponding author

Received: March 31, 2020

Revised: July 31, 2020

Early access: October 2020

Published: May 2021

The first author is supported by DFG grant GR 4899/1-1. The second author is supported by FWF Project P31950-N35

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Abstract

We consider a minimal equicontinuous action of a finitely generated group $ G $ on a Cantor set $ X $ with invariant probability measure $ \mu $, and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $ H $ of $ G $ such that the set of points in $ X $ whose stabilizers are conjugate to $ H $ has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.

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Mathematics Subject Classification: Primary: 37B05, 37A15, 22F10; Secondary: 22F50, 37E25.

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