论文标题
简单的瓷砖和吸引子
Simple tiles and attractors
论文作者
论文摘要
我们在空间中研究自相似的吸引子$ \ mathbb {r}^d $，即由几个具有相同线性部分的仿射操作员定义的自相似紧凑型集合。当仿射操作员的线性部分的矩阵$ m $和变化是整数时，吸引子的特殊情况在文献中是众所周知的，这是由于小波构建和近似理论的许多应用而闻名。在这种情况下，如果吸引子的测量值，则称为瓷砖。如果它们是Polyhedra或有限的许多Polyhedra联合，我们将自相似的吸引子和瓷砖分类。我们获得了整数收缩矩阵的完整描述以及瓷砖合理教的数字集和以任意维度为单位的凸形图块的完整描述。事实证明，在二维平面上，每个多边形瓷砖（不一定是凸）必须是平行四边形。给出了多维瓷砖的非平凡示例，这些瓷砖是有限的Polyhedra联合的，在此情况下，提供了完整的分类。考虑到$ \ mathbb {r}^d $和整数单变量瓷砖中的正常HAAR系统的应用。
We study self-similar attractors in the space $\mathbb{R}^d$, i.e., self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix $M$ of the linear part of affine operators and the shifts are integer, is well known in the literature due to many applications in the construction of wavelet and in approximation theory. In this case, if an attractor has measure one, it is called a tile. We classify self-similar attractors and tiles in case when they are either polyhedra or union of finitely many polyhedra. We obtain a complete description of the integer contraction matrices and of the digit sets for tiles-parallelepipeds and for convex tiles in arbitrary dimension. It is proved that on a two-dimensional plane, every polygonal tile (not necessarily convex) must be a parallelogram. Non-trivial examples of multidimensional tiles which are a finite union of polyhedra are given, and in the case $d = 1$ their complete classification is provided. Applications to orthonormal Haar systems in $\mathbb{R}^d$ and to integer univariate tiles are considered.
