论文标题
二分法，用于DeLono集合的界位置等效性
A dichotomy for bounded displacement equivalence of Delone sets
论文作者
论文摘要
我们证明，在$ \ mathbb {r}^d $中，在DeLone集的每个紧凑型空间中，对于通过翻译的动作而言，这是最小的，要么所有Delone集都均匀地传播，要么连续许多不同的有界位移等价类都代表了，没有一个包含晶格。隐含的限制是相对于chabauty- topology采取的，这是$ \ mathbb {r}^d $的封闭子集的自然拓扑。这种拓扑与有限的局部复杂性设置中的标准局部拓扑结合在一起，因此，二分法均适用于所有与认识良好的结构相关的DeLono集合的最小空间，例如剪切和项目的替代集和替代瓷砖，无论是否假定有限的局部复杂性。
We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of $\mathbb{R}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.
