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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Irregular model sets and tame dynamics


Authors: G. Fuhrmann, E. Glasner, T. Jäger and C. Oertel
Journal: Trans. Amer. Math. Soc. 374 (2021), 3703-3734
MSC (2020): Primary 52C23; Secondary 37B10, 37B40
DOI: https://doi.org/10.1090/tran/8349
Published electronically: March 8, 2021
MathSciNet review: 4237960
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Abstract:

We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces.

In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting.


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Additional Information

G. Fuhrmann
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
MR Author ID: 1149230
ORCID: 0000-0002-0634-0802
Email: gabriel.fuhrmann@durham.ac.uk

E. Glasner
Affiliation: Department of Mathematics, Tel-Aviv University, Ramat Aviv, Israel
MR Author ID: 271825
ORCID: 0000-0003-1167-1283
Email: glasner@math.tau.ac.il

T. Jäger
Affiliation: Institute of Mathematics, Friedrich Schiller University Jena, Germany
Email: tobias.jaeger@uni-jena.de

C. Oertel
Affiliation: Institute of Mathematics, Friedrich Schiller University Jena, Germany
MR Author ID: 1360927
Email: christian.oertel@uni-jena.de

Keywords: Model sets, cut and project schemes, topological group actions, tame dynamics
Received by editor(s): June 18, 2019
Received by editor(s) in revised form: June 12, 2020, and October 6, 2020
Published electronically: March 8, 2021
Additional Notes: The third author was supported by a Heisenberg professorship of the German Research Council (DFG grant OE 538/6-1). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 750865
Article copyright: © Copyright 2021 American Mathematical Society