A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a Pisot–Vijayaraghavan (PV) number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

中文翻译：

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一系列最小且可重整化的矩形交换映射

一种域名交换图(DEM) 是定义在平滑 Jordan 域上的动态系统，它是分段平移。我们解释了如何使用剪切和投影集来构建最小 DEM。专门针对域是正方形并且切割和投影集与 Galois 格相关联的情况，我们构建了一个无限的 DEM 系列，其中每个映射都与 Pisot-Vijayaraghavan (PV) 数相关联。我们为这些 DEM 开发了一个重整化方案。该系列中的某些 DEM 可以组合成多级、可重整化的 DEM。