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Number-theoretic positive entropy shifts with small centralizer and large normalizer

Part of: Discrete geometry Geometry of numbers Topological dynamics Classical measure theory

Published online by Cambridge University Press:  04 November 2020

M. BAAKE ,
Á. BUSTOS ,
C. HUCK ,
M. LEMAŃCZYK  and
A. NICKEL
M. BAAKE*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501Bielefeld, Germany (e-mail: abustos,huck@math.uni-bielefeld.de)
Á. BUSTOS
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501Bielefeld, Germany (e-mail: abustos,huck@math.uni-bielefeld.de)
C. HUCK
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501Bielefeld, Germany (e-mail: abustos,huck@math.uni-bielefeld.de)
M. LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin Street, 87-100 Toruń, Poland (e-mail: mlem@mat.umk.pl)
A. NICKEL
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127Essen, Germany (e-mail: andreas.nickel@uni-due.de)
*
e-mail: mbaake@math.uni-bielefeld.de
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Abstract

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Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

Keywords

number-theoretic shift spacestopological centralizer and normalizerk-free integers in number fields

MSC classification

Primary: 37B10: Symbolic dynamics 11H06: Lattices and convex bodies
Secondary: 52C23: Quasicrystals, aperiodic tilings 28A80: Fractals
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3201 - 3226
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Apostol, T. M.. Introduction to Analytic Number Theory, corr. 4th printing. Springer, New York, 1984.Google Scholar
Baake, M.. A brief guide to reversing and extended symmetries of dynamical systems. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Ed. Ferenczi, S., Kułaga-Przymus, J. and Lemańczyk, M.. Springer, Cham, 2018, pp. 117–135.CrossRefGoogle Scholar
Baake, M., Bustos, Á. and Nickel, A.. On the stabiliser of some number-theoretic shift spaces, in preparation.Google Scholar
Baake, M. and Grimm, U.. Aperiodic Order. Vol. 1. A Mathematical Invitation. Cambridge University Press, Cambridge, 2013.Google Scholar
Baake, M. and Huck, C.. Ergodic properties of visible lattice points. Proc. Steklov Inst. Math. 288 (2015), 184–208.CrossRefGoogle Scholar
Baake, M., Huck, C. and Strungaru, N.. On weak model sets of extremal density. Indag. Math. 28 (2017), 3–31.CrossRefGoogle Scholar
Baake, M. and Lenz, D.. Dynamical systems on translation bounded measure: pure point dynamical and diffraction spectra. Ergodic Th. & Dynam. Sys. 24 (2004), 1867–1893.CrossRefGoogle Scholar
Baake, M., Moody, R. V. and Pleasants, P. A. B.. Diffraction of visible lattice points and $k$ th power free integers. Discrete Math. 221 (2000), 3–42.CrossRefGoogle Scholar
Baake, M., Roberts, J. A. G. and Yassawi, R.. Reversing and extended symmetries of shift spaces. Discrete Contin. Dyn. Syst. A 38 (2018), 835–866.CrossRefGoogle Scholar
Baake, M., Spindeler, T. and Strungaru., N. Diffraction of compatible random substitutions in one dimension. Indag. Math. 29 (2018), 1031–1071CrossRefGoogle Scholar
Blomer, V.. Private communication, 2017.Google Scholar
Borevich, Z. I. and Shafarevich., I. R. Number Theory. Academic Press, New York, 1966.Google Scholar
Bustos., Á. Extended symmetry groups of multidimensional subshifts with hierarchical structure. Discrete Contin. Dyn. Syst. A 40 (2020), 5869–5895.CrossRefGoogle Scholar
Cellarosi, F. and Vinogradov., I. Ergodic properties of $k$ -free integers in number fields. J. Mod. Dyn. 7 (2013), 461–488.CrossRefGoogle Scholar
Cortez, M. I. and Petite., S. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete Contin. Dyn. Syst. A 40 (2020), 2891–2901.CrossRefGoogle Scholar
Coven, E. M., Quas, A. and Yassawi., R. Computing automorphism groups of shifts using atypical equivalence classes. Discrete Anal. 3 (2016), 1–24.Google Scholar
Cyr, V. and Kra., B. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma 3 (2015), e5:1–27.CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite., S. On automorphism groups of low complexity subshifts. Ergodic Th. & Dynam. Sys. 36 (2016), 64–95.CrossRefGoogle Scholar
Dymek, A., Kasjan, S., Kułaga-Przymus, J. and Lemańczyk, M.. ${\mathcal{B}}$ -free sets and dynamics. Trans. Amer. Math. Soc. 370 (2018), 5425–5489.CrossRefGoogle Scholar
El Abdalaoui, E. H., Lemańczyk, M. and de la Rue, T.. A dynamical point of view on the set of ${\mathcal{B}}$ -free integers. Int. Math. Res. Not. 16 (2015), 7258–7286.CrossRefGoogle Scholar
Goodson, G., del Junco, A., Lemańczyk, M. and Rudolph., D. Ergodic transformations conjugate to their inverses by involutions. Ergodic Th. & Dynam. Sys. 16 (1996), 97–124.CrossRefGoogle Scholar
Hardy, G. M. and Wright., E. M. An Introduction to the Theory of Numbers, 6th edn. Ed. Heath-Brown, D. R. and Silverman, J. H.. Oxford University Press, Oxford, 2008.Google Scholar
Kasjan, S., Keller, G. and Lemańczyk., M. Dynamics of ${\mathcal{B}}$ -free sets: a view through the window. Int. Math. Res. Not. 9 (2019), 2690–2734.CrossRefGoogle Scholar
Keller., G. Maximal equicontinuous generic factors and weak model sets. Discrete Contin. Dyn. Syst. A, 40 (2020), 6855–6875.CrossRefGoogle Scholar
Keller., G. Private communication, 2019.Google Scholar
Kim, Y.-O., Lee, J. and Park, K. K.. A zeta function for flip systems. Pacific J. Math. 209 (2003), 289–301.CrossRefGoogle Scholar
Kitchens, B. and Schmidt, K.. Isomorphism rigidity of irreducible algebraic ${\mathbb{Z}}^d$ -actions. Invent. Math. 142 (2000), 559–577.CrossRefGoogle Scholar
Lind, D. and Marcus., B. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, New York, 1995.CrossRefGoogle Scholar
Mentzen., M. K. Automorphisms of subshifts defined by ${\mathcal{B}}$ -free sets of integers. Colloq. Math. 147 (2017), 87–94.CrossRefGoogle Scholar
Neukirch., J. Algebraic Number Theory. Springer, Berlin, 1999.CrossRefGoogle Scholar
O’Farrel, A. G. and Short., I. Reversibility in Dynamics and Group Theory. Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Pleasants, P. A. B. and Huck., C. Entropy and diffraction of the $k$ -free points in $n$ -dimensional lattices. Discrete Comput. Geom. 50 (2013), 39–68.CrossRefGoogle Scholar
Roberts, J. A. G. and Quispel., G. R. W. Chaos and time-reversal symmetry: order and chaos in reversible dynamical systems. Phys. Rep. 216 (1992), 63–177.CrossRefGoogle Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin. Birkhäuser, Basel, 1995.Google Scholar
Zagier, D.. Zetafunktionen und quadratische Körper. Springer, Berlin, 1981.CrossRefGoogle Scholar