Cut and project sets with polytopal window II: Linear repetitivity

Henna Koivusalo, James J. Walton
{"title":"Cut and project sets with polytopal window II: Linear repetitivity","authors":"Henna Koivusalo, James J. Walton","doi":"10.1090/tran/8633","DOIUrl":null,"url":null,"abstract":"This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice projects densely into the internal space and (possibly after translation) hits each supporting hyperplane of the polytopal window. Our main result is that LR is satisfied if and only if the patterns are of low complexity (property C), and the projected lattice satisfies a Diophantine condition (property D). Property C can be checked by computation of the ranks and dimensions of linear spans of the stabiliser subgroups of the supporting hyperplanes, as investigated in Part I to this article. To define the correct Diophantine condition D, we establish new results on decomposing polytopal cut and project schemes to factors, developing concepts initiated in the work of Forrest, Hunton and Kellendonk. This means that, when C is satisfied, the window splits into components which induce a compatible splitting of the lattice. Then property D is the requirement that, for any suitable decomposition, these factors do not project close to the origin in the internal space, relative to the norm in the total space. On each factor, this corresponds to the usual notion from Diophantine Approximation of a system of linear forms being badly approximable. This extends previous work on cubical cut and project schemes to a very general class of cut and project schemes. We demonstrate our main theorem on several examples, and derive some further consequences of our main theorem, such as the equivalence LR, positivity of weights and satisfying a subadditive ergodic theorem for this class of polytopal cut and project sets.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"11 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Dynamical Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tran/8633","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

Abstract

This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice projects densely into the internal space and (possibly after translation) hits each supporting hyperplane of the polytopal window. Our main result is that LR is satisfied if and only if the patterns are of low complexity (property C), and the projected lattice satisfies a Diophantine condition (property D). Property C can be checked by computation of the ranks and dimensions of linear spans of the stabiliser subgroups of the supporting hyperplanes, as investigated in Part I to this article. To define the correct Diophantine condition D, we establish new results on decomposing polytopal cut and project schemes to factors, developing concepts initiated in the work of Forrest, Hunton and Kellendonk. This means that, when C is satisfied, the window splits into components which induce a compatible splitting of the lattice. Then property D is the requirement that, for any suitable decomposition, these factors do not project close to the origin in the internal space, relative to the norm in the total space. On each factor, this corresponds to the usual notion from Diophantine Approximation of a system of linear forms being badly approximable. This extends previous work on cubical cut and project schemes to a very general class of cut and project schemes. We demonstrate our main theorem on several examples, and derive some further consequences of our main theorem, such as the equivalence LR, positivity of weights and satisfying a subadditive ergodic theorem for this class of polytopal cut and project sets.
剪切和项目设置多边形窗口II:线性重复
本文给出了一类自然的具有凸多边形窗口的非周期欧几里得切割投影格式的线性重复性的完全分类。我们的结果涵盖了那些切割和投影方案,其中晶格密集地投影到内部空间并且(可能在平移之后)击中多边形窗口的每个支持超平面。我们的主要结果是,当且仅当模式是低复杂度的(性质C),并且投影格满足Diophantine条件(性质D)时,LR是满足的。性质C可以通过计算支撑超平面的稳定子群的线性跨度的秩和尺寸来检验,如本文第一部分所研究的那样。为了定义正确的丢番图条件D,我们建立了分解多面体切割和项目方案的新结果,发展了Forrest, Hunton和Kellendonk工作中提出的概念。这意味着,当C满足时,窗口分裂成组件,从而诱导晶格的兼容分裂。那么性质D是要求,对于任何合适的分解,这些因子在内部空间中的投影不靠近原点,相对于总空间中的范数。在每一个因素上,这都对应于丢番图近似中关于线性形式的系统难以近似的通常概念。这将以前关于立方体切割和项目方案的工作扩展到非常一般的切割和项目方案类。我们在几个例子上证明了我们的主要定理,并得到了我们的主要定理的一些进一步的结果,如等价LR,权的正性和满足这类多边形切集和投影集的次加性遍历定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信