论文标题
多尺度替代瓷砖
Multiscale Substitution Tilings
论文作者
论文摘要
我们介绍了一个新的通用框架，用于构建欧几里得空间的瓷砖，我们称之为多尺度替代瓷砖。这些整理是由有限的原始原始替代方案产生的，其中允许使用多个不同的缩放常数。这与经过良好的替代瓷砖的标准情况相反，该替代方案包括penrose和风车砖等示例。在缩放常数的额外非理性假设下，我们的构造定义了一类新的瓷砖和平铺空间，这些空间与标准设置中出现的瓷砖空间本质上不同。我们研究这些新对象的各种结构，几何，统计和动力学方面，并建立了各种各样的特性。我们的主要结果包括显式密度公式和相关瓷砖动力学系统的独特奇迹性。
We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well-studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from those that arise in the standard setup. We study various structural, geometric, statistical and dynamical aspects of these new objects and establish a wide variety of properties. Among our main results are explicit density formulas and the unique ergodicity of the associated tiling dynamical systems.
