Geometry of canonical self-similar tilings
Geometry of canonical self-similar tilings
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规范自相似平铺的几何
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
S. Winter
中科院分区：
文献类型：
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作者：
Erin P. J. Pearse;S. Winter
本文给出了自相似集F的平行集F_n可由相应的正则镶嵌T的内平行集T_{-T}在点{SST}意义下描述的情形的几种不同的几何刻画。例如，$F_n =T_{-n} cup C_n $当且仅当$F$的凸船体$C$的边界是$F$的子集，或者如果$E$的边界，即$F$的补的无界部分，是凸集的边界。在特征化的情况下，平铺允许获得$F$的管公式，即，作为$F_n $的函数的$F_n $体积的表达式。在此基础上，我们进一步阐明了正则镶嵌的一些几何性质。
出于寻找管公式，我们给出了一个推广的平铺结构，适用于所有的自仿射集F$有空的内部和满足开集条件。我们还刻画了$F$的平行集和这些平铺之间的关系。
We give several different geometric characterizations of the situation in which the parallel set $F_epsilon$ of a self-similar set $F$ can be described by the inner $epsilon$-parallel set $T_{-epsilon}$ of the associated canonical tiling $mathcal T$, in the sense of cite{SST}. For example, $F_epsilon=T_{-epsilon} cup C_epsilon$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_epsilon$ as a function of $epsilon$. On the way, we clarify some geometric properties of canonical tilings.
Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings.