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Orbit separation dimension as complexity measure for primitive inflation tilings

Part of: Discrete geometry

Published online by Cambridge University Press:  13 May 2025

MICHAEL BAAKE Open the ORCID record for MICHAEL BAAKE [Opens in a new window] ,
FRANZ GÄHLER Open the ORCID record for FRANZ GÄHLER [Opens in a new window]  and
PHILIPP GOHLKE Open the ORCID record for PHILIPP GOHLKE [Opens in a new window]
MICHAEL BAAKE*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (e-mail: gaehler@math.uni-bielefeld.de)
FRANZ GÄHLER
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (e-mail: gaehler@math.uni-bielefeld.de)
PHILIPP GOHLKE
Affiliation:
Lund University, Centre for Mathematical Sciences, Box 118, 221 00 Lund, Sweden Institut für Mathematik, Universität Jena, Ernst-Abbe-Platz 1–2, 07743 Jena, Germany (e-mail: philipp.gohlke@uni-jena.de)
*
e-mail: mbaake@math.uni-bielefeld.de
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Abstract

Orbit separation dimension ($\mathrm {OSD}$), previously introduced as amorphic complexity, is a powerful complexity measure for topological dynamical systems with pure-point spectrum. Here, we develop methods and tools for it that allow a systematic application to translation dynamical systems of tiling spaces that are generated by primitive inflation rules. These systems share many nice properties that permit the explicit computation of the $\mathrm {OSD}$, thus providing a rich class of examples with non-trivial $\mathrm {OSD}$.

Keywords

inflation tilingstopological dynamicsinvariantscomplexity

MSC classification

Secondary: 52C23: Quasicrystals, aperiodic tilings

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 10 , October 2025 , pp. 2992 - 3020
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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