Admissible Reversing and Extended Symmetries for Bijective Substitutions

被引:0
作者
Álvaro Bustos
Daniel Luz
Neil Mañibo
机构
[1] Departamento de Ingeniería Matemática Universidad de Chile,Fakultät für Mathematik
[2] Universität Bielefeld,undefined
来源
Discrete & Computational Geometry | 2023年 / 69卷
关键词
Extended symmetries; Automorphism groups; Substitution subshifts; Aperiodic tilings; 52C23; 37B10; 37B52; 20B27;
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学科分类号
摘要
In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to compute them. Moreover, for any finite group H and any subgroup P of the d-dimensional hyperoctahedral group, we construct a bijective substitution which generates an aperiodic subshift with symmetry group Zd×H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^{d}\times H$$\end{document} and extended symmetry group (Zd⋊P)×H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {Z}}^{d} \rtimes P)\times H$$\end{document}. A similar construction with the same symmetry group, but with extended symmetry group (Zd×H)⋊P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {Z}}^{d} \times H) \rtimes P$$\end{document} is also provided under a mild assumption on the dimension.
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页码:800 / 833
页数:33
相关论文
共 48 条
[1]  
Baake M(1984)Structure and representations of the hyperoctahedral group J. Math. Phys. 25 3171-3182
[2]  
Baake M(2021)Number-theoretic positive entropy shifts with small centralizer and large normalizer Ergodic Theory Dyn. Syst. 41 3201-3226
[3]  
Bustos Á(2006)The structure of reversing symmetry groups Bull. Aust. Math. Soc. 73 445-459
[4]  
Huck C(2018)Reversing and extended symmetries of shift spaces Discrete Contin. Dyn. Syst. 38 835-866
[5]  
Lemańczyk M(2018)Spectral theory of Ergodic Theory Dyn. Syst. 38 1289-1341
[6]  
Nickel A(2008) substitutions Colloq. Math. 110 409-429
[7]  
Baake M(1988)Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems Trans. Am. Math. Soc. 306 71-114
[8]  
Roberts JAG(2020)The automorphism group of a shift of finite type Discrete Contin. Dyn. Syst. 40 5869-5895
[9]  
Baake M(2008)Extended symmetry groups of multidimensional subshifts with hierarchical structure Discrete Comput. Geom. 40 622-640
[10]  
Roberts JAG(2020)Self-similar tiling systems, topological factors and stretching factors Discrete Contin. Dyn. Syst. 40 2891-2901