| Abstract: |
| After the discovery of quasicrystals in material sciences, there has been a lot of study to understand the structure of quasicrystals. Mathematically, quasicrystals can be modeled by tilings or point sets, and the structure of quasicrystals can be described by pure discrete spectrum of the tiling dynamics. It is known that a cut-and-project set with a nice window always gives pure discrete spectrum. But the converse is not true in general settings. Here we look at substitution tilings and study the relation between the cut-and-project sets and pure discrete spectrum. We first look at the case that the expansion maps of the substitutions are diagonalizable. And then we will also talk about a recent development on non-diagonalizable case. |
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