Hostname: page-component-5b777bbd6c-f9nfp Total loading time: 0 Render date: 2025-06-19T21:59:13.665Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Akiyama, S. and Araki, Y.. An alternative proof for an aperiodic monotile. Discrete Comput. Geom., to appear; arXiv:2207.12322.Google Scholar
Akiyama, S., Gähler, F. and Lee, J.-Y.. Determining pure discrete spectrum for some self-affine tilings. Discrete Math. Theor. Comput. Sci. 163 (2014), 305–316.Google Scholar
Akiyama, S. and Lee, J.-Y.. Algorithm for determining pure pointedness of self-affine tilings. Adv. Math. 226 (2011), 2855–2883.CrossRefGoogle Scholar
Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated ${C}^{\ast }$ -algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509–537.CrossRefGoogle Scholar
Aujogue, J.-B., Barge, M., Kellendonk, J. and Lenz, D.. Equicontinuous factors, proximality and Ellis semigroup for Delone sets. Mathematics of Aperiodic Order. Ed. Kellendonk, J., Lenz, D. and Savinien, J.. Birkhäuser, Basel, 2015, pp. 137–194.CrossRefGoogle Scholar
Auslander, J.. Minimal Flows and their Extensions. North Holland, Amsterdam, 1988.Google Scholar
Baake, M., Gähler, F. and Sadun, L.. Dynamics and topology of the Hat family of tilings. Israel J. Math., to appear.Google Scholar
Baake, M., Frank, N. P. and Grimm, U.. Three variations on a theme by Fibonacci. Stoch. Dyn. 21 (2021), 2140001.CrossRefGoogle Scholar
Baake, M., Gähler, F. and Mañibo, N.. Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction. Comm. Math. Phys. 370 (2019), 591–635.CrossRefGoogle Scholar
Baake, M., Gähler, F. and Mazáč, J.. Fibonacci direct product variation tilings. J. Math. Phys. 63 (2022) 082702:1–13.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Aperiodic Order. Volume 1: A Mathematical Invitation. Cambridge University Press, Cambridge, 2013.CrossRefGoogle Scholar
Baake, M. and Grimm, U.. Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings. Doc. Math. 25 (2020), 2303–2337.Google Scholar
Baake, M. and Moody, R. V.. Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. (Crelle) 573 (2004), 61–94.Google Scholar
Barge, M. and Diamond, B.. Proximality in Pisot tiling spaces. Fund. Math. 194 (2007), 191–238.CrossRefGoogle Scholar
Barge, M. and Kellendonk, J.. Proximality and pure point spectrum for tiling dynamical systems. Michigan Math. J. 62 (2013), 793–822.CrossRefGoogle Scholar
Bruin, H.. Topological and Ergodic Theory of Symbolic Dynamics. American Mathematical Society, Providence, RI, 2022.Google Scholar
Clark, A. and Sadun, L.. When shape matters. Ergod. Th. & Dynam. Sys. 26 (2006), 69–86.CrossRefGoogle Scholar
Das, M. and Edgar, G. A.. Separation properties for graph-directed self-similar fractals. Topology Appl. 152 (2005), 138–156.CrossRefGoogle Scholar
Edgar, G. A. and Golds, J.. A fractal dimension estimate for a graph-directed iterated function system of non-similarities. Indiana Univ. Math. J. 48 (1999), 429–447.CrossRefGoogle Scholar
Feng, D.-J., Furukado, M., Ito, S. and Wu, J., Pisot substitutions and the Hausdorff dimension of boundaries of atomic surfaces. Tsukuba J. Math. 30 (2006), 195–223.CrossRefGoogle Scholar
Fuhrmann, G. and Gröger, M.. Constant length substitutions, iterated function systems and amorphic complexity. Math. Z. 295 (2020), 1385–1404.CrossRefGoogle Scholar
Fuhrmann, G., Gröger, M. and Jäger, T.. Amorphic complexity. Nonlinearity 29 (2016), 528–565.CrossRefGoogle Scholar
Fuhrmann, G., Gröger, M., Jäger, T. and Kwietniak, D.. Amorphic complexity of group actions with applications to quasicrystals. Trans. Amer. Math. Soc. 376 (2023), 2395–2418.Google Scholar
Fuhrmann, G., Gröger, M. and Lenz, D.. The structure of mean equicontinuous group actions. Israel J. Math. 247 (2022), 75–123.CrossRefGoogle Scholar
Kellendonk, J.. Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7 (1995), 1133–1180.CrossRefGoogle Scholar
Kenyon, R.. Inflationary tilings with a similarity structure, Comment. Math. Helv. 69 (1994), 169–198.CrossRefGoogle Scholar
Lee, J.-Y.. Pure point diffractive substitution Delone sets have the Meyer property. Discrete Comput. Geom. 39 (2008), 219–338.CrossRefGoogle Scholar
Lee, J.-Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3 (2002), 1003–1018.CrossRefGoogle Scholar
Lee, J.-Y. and Solomyak, B.. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete Contin. Dyn. Syst. Ser. A 32 (2012), 935–959.CrossRefGoogle Scholar
Luck, J. M., Godrèche, C., Janner, A. and Janssen, T.. The nature of the atomic surfaces of quasiperiodic self-similar structures. J. Phys. A 26 (1993), 1951–1999.CrossRefGoogle Scholar
Maloney, G. R. and Rust, D.. Beyond primitivity for one-dimensional substitution subshifts and tiling spaces. Ergod. Th. & Dynam. Sys. 38 (2018), 1086–1117.CrossRefGoogle Scholar
Petersen, K.. Measuring Complexity in Cantor Dynamics. Unpublished lectures notes (CIMPA Research School, 2015); Preprint, 2016, arXiv:1607.02425.Google Scholar
Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Ed. Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A.. Springer, Berlin, 2002.CrossRefGoogle Scholar
Sing, B.. Pisot substitutions and beyond. PhD Thesis, Bielefeld University, 2007; available electronically at urn:nbn:de:hbz:361-11555.Google Scholar
Sirvent, V. F. and Solomyak, B.. Pure discrete spectrum for one-dimensional substitution systems of Pisot type. Canad. Math. Bull. 45 (2002), 697–710.CrossRefGoogle Scholar
Smith, D., Myers, J. S., Kaplan, C. S. and Goodman-Strauss, C.. An aperiodic monotile. Combin. Theory 4 (2024), 6, 1–90.Google Scholar
Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17 (1997), 695–738; Ergod. Th. & Dynam. Sys. 19 (1999), 1685 (erratum).CrossRefGoogle Scholar
Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265–279.CrossRefGoogle Scholar
Solomyak, B.. Eigenfunctions for substitution tiling systems. Adv. Stud. Pure Math. 49 (2007), 433–454.CrossRefGoogle Scholar
Strungaru, N.. Almost periodic pure point measures. Aperiodic Order. Volume 2: Crystallography and Almost Periodicity. Ed. Baake, M. and Grimm, U.. Cambridge University Press, Cambridge, 2017, pp. 271–342.CrossRefGoogle Scholar