Amorphic complexity of group actions with applications to quasicrystals

Fuhrmann, Gabriel; Gröger, Maik; Jäger, Tobias; Kwietniak, Dominik

doi:10.1090/tran/8700

Amorphic complexity of group actions with applications to quasicrystals
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by Gabriel Fuhrmann, Maik Gröger, Tobias Jäger and Dominik Kwietniak;

Trans. Amer. Math. Soc. 376 (2023), 2395-2418

DOI: https://doi.org/10.1090/tran/8700

Published electronically: January 18, 2023

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Abstract:

In this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb {Z}$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer’s cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.

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Bibliographic Information

Gabriel Fuhrmann
Affiliation: Department of Mathematical Sciences, Durham University, United Kingdom
MR Author ID: 1149230
ORCID: 0000-0002-0634-0802
Email: gabriel.fuhrmann@durham.ac.uk
Maik Gröger
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
ORCID: 0000-0003-0802-8932
Email: maik.groeger@im.uj.edu.pl
Tobias Jäger
Affiliation: Department of Mathematics, University of Jena, Germany
ORCID: 0000-0001-8457-8392
Email: tobias.jaeger@uni-jena.de
Dominik Kwietniak
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
MR Author ID: 773622
ORCID: 0000-0002-7794-2835
Email: dominik.kwietniak@uj.edu.pl
Received by editor(s): May 14, 2021
Received by editor(s) in revised form: January 17, 2022, and February 23, 2022
Published electronically: January 18, 2023
Additional Notes: This project received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 750865. The research leading to these results received funding from the Norwegian Financial Mechanism 2014-2021 via the POLS grant no. 2020/37/K/ST1/02770. Furthermore, it received support by the DFG Emmy-Noether grant Ja 1721/2-1 and DFG Heisenberg grant Oe 538/6-1.
The fourth author was supported by the National Science Centre, Poland, grant no. 2018/29/B/ST1/01340.
Journal: Trans. Amer. Math. Soc. 376 (2023), 2395-2418
DOI: https://doi.org/10.1090/tran/8700
MathSciNet review: 4557869