Cut-and-Project Schemes for Pisot Family Substitution Tilings
1
Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea, or KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
2
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan
3
Lehrstuhl für Mathematik und Statistik, Department Mathematik und Informationstechnologie, Montanuniversität, Franz Josef Strasse 18, A-8700 Leoben, Austria
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 511; https://doi.org/10.3390/sym10100511
Received: 19 September 2018 / Revised: 9 October 2018 / Accepted: 11 October 2018 / Published: 16 October 2018
We consider Pisot family substitution tilings in 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 for some integer 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 , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.
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Lee, J.-Y.; Akiyama, S.; Nagai, Y. Cut-and-Project Schemes for Pisot Family Substitution Tilings. Symmetry 2018, 10, 511.
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