We study self-similar attractors in the space , that is, self- similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix of the linear part and the shifts of the affine operators are integer, is well known in the literature due to the many applications in the theory of wavelets 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 the case when they are either polyhedra or a union of finitely many polyhedra. We obtain a complete description of the matrices and the digit sets for parallelepiped tiles and for convex tiles in arbitrary dimensions. It is proved that on a two-dimensional plane, every polygonal tile (not necessarily convex) must be a parallelogram. Nontrivial examples of multidimensional tiles which are a finite union of polyhedra are given, and in the case a complete classification is provided for them. Applications to orthonormal Haar systems in and to integer univariate tiles are considered.

Bibliography: 18 titles.

我们研究空间中的自相似吸引子，即由具有相同线性部分的多个仿射算子定义的自相似紧集。当线性部分的矩阵和仿射算子的位移为整数时，吸引子的特殊情况在文献中是众所周知的，这是由于在小波理论和逼近理论中有许多应用。在这种情况下，如果一个吸引子有一个小节，则称为平铺。当自相似吸引子和瓦片为多面体或有限多个多面体的并集时，我们将它们分类。我们获得了矩阵的完整描述以及任意尺寸的平行六面体瓷砖和凸面瓷砖的数字集。事实证明，在二维平面上，每个多边形图块（不一定是凸面）都必须是平行四边形。给出了多面体的有限并集的多维平铺的非平凡示例，并且在这种情况下为它们提供了完整的分类。考虑将其应用于正交单变量瓦片中的正交Haar系统。

参考书目：18种。