Realization of big centralizers of minimal aperiodic actions on the Cantor set

María Isabel Cortez; Samuel Petite

doi:10.3934/dcds.2020153

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2020, Volume 40, Issue 5: 2891-2901. Doi: 10.3934/dcds.2020153

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Realization of big centralizers of minimal aperiodic actions on the Cantor set

María Isabel Cortez^1, and
Samuel Petite^2,

1.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Avenida Libertador Bernardo O'Higgins 3363, Santiago, Chile
2.
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS-UMR 7352, Université de Picardie Jules Verne, 33, rue Saint Leu 80039 Amiens Cedex 1, France

Received: July 31, 2019

Revised: October 31, 2019

Published: March 2020

The first author was supported by proyecto Fondecyt Regular No. 1190538.This research was supported through the cooperation project MathAmSud DCS 38889 TM

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Abstract

In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $ \mathbb Z $ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $ G $, the centralizer of any minimal aperiodic $ G $-action on the Cantor set is a subgroup of the centralizer of a minimal $ \mathbb Z $-action.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 22F50.

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