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Dynamical simplices and minimal homeomorphisms


Authors: Tomás Ibarlucía and Julien Melleray
Journal: Proc. Amer. Math. Soc. 145 (2017), 4981-4994
MSC (2010): Primary 54H20; Secondary 37B05
DOI: https://doi.org/10.1090/proc/13578
Published electronically: April 4, 2017
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Abstract: We give a characterization of sets $ K$ of probability measures on a Cantor space $ X$ with the property that there exists a minimal homeomorphism $ g$ of $ X$ such that the set of $ g$-invariant probability measures on $ X$ coincides with $ K$. This extends theorems of Akin (corresponding to the case when $ K$ is a singleton) and Dahl (when $ K$ is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's.


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Additional Information

Tomás Ibarlucía
Affiliation: Université de Lyon, Université Claude Bernard – Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Julien Melleray
Affiliation: Université de Lyon, Université Claude Bernard – Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
Address at time of publication: Institut de Math’ematiques de Jussieu–PRG, Université Paris 7, case 7012, 75205 Parist cedex 13, France

DOI: https://doi.org/10.1090/proc/13578
Keywords: Cantor dynamics, minimal homeomorphism, dynamical simplex, invariant measures, Kakutani-Rokhlin partitions
Received by editor(s): December 11, 2015
Received by editor(s) in revised form: October 28, 2016
Published electronically: April 4, 2017
Additional Notes: Research partially supported by Agence Nationale de la Recherche projects GruPoLoCo (ANR-11-JS01-0008) and ValCoMo (ANR-13-BS01-0006).
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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