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

Part of:Classical measure theoryTopological dynamicsGeometry of numbersDiscrete geometry

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, 33501 Bielefeld, Germany (e-mail: abustos,huck@math.uni-bielefeld.de)
Á. BUSTOS
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (e-mail: abustos,huck@math.uni-bielefeld.de)
C. HUCK
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, 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, 45127 Essen, 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 dynamics11H06: Lattices and convex bodies
Secondary:52C23: Quasicrystals, aperiodic tilings28A80: Fractals

Information

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

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