Cut-and-Project Schemes for Pisot Family Substitution Tilings
We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m...
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MDPI AG
2018-10-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/10/10/511 |
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| author | Jeong-Yup Lee Shigeki Akiyama Yasushi Nagai |
| author_facet | Jeong-Yup Lee Shigeki Akiyama Yasushi Nagai |
| author_sort | Jeong-Yup Lee |
| collection | DOAJ |
| description | We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz. |
| first_indexed | 2024-04-13T00:27:04Z |
| format | Article |
| id | doaj.art-bd8de2b45fcb4c15a1b8b0a29a2de45c |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T00:27:04Z |
| publishDate | 2018-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bd8de2b45fcb4c15a1b8b0a29a2de45c2022-12-22T03:10:35ZengMDPI AGSymmetry2073-89942018-10-01101051110.3390/sym10100511sym10100511Cut-and-Project Schemes for Pisot Family Substitution TilingsJeong-Yup Lee0Shigeki Akiyama1Yasushi Nagai2Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea, or KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, KoreaInstitute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, JapanLehrstuhl für Mathematik und Statistik, Department Mathematik und Informationstechnologie, Montanuniversität, Franz Josef Strasse 18, A-8700 Leoben, AustriaWe consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.http://www.mdpi.com/2073-8994/10/10/511Pisot substitution tilingspure point spectrumregular model setalgebraic coincidence |
| spellingShingle | Jeong-Yup Lee Shigeki Akiyama Yasushi Nagai Cut-and-Project Schemes for Pisot Family Substitution Tilings Symmetry Pisot substitution tilings pure point spectrum regular model set algebraic coincidence |
| title | Cut-and-Project Schemes for Pisot Family Substitution Tilings |
| title_full | Cut-and-Project Schemes for Pisot Family Substitution Tilings |
| title_fullStr | Cut-and-Project Schemes for Pisot Family Substitution Tilings |
| title_full_unstemmed | Cut-and-Project Schemes for Pisot Family Substitution Tilings |
| title_short | Cut-and-Project Schemes for Pisot Family Substitution Tilings |
| title_sort | cut and project schemes for pisot family substitution tilings |
| topic | Pisot substitution tilings pure point spectrum regular model set algebraic coincidence |
| url | http://www.mdpi.com/2073-8994/10/10/511 |
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