Simple tiles and attractors

We study self-similar attractors in the space $\mathbb R^d$, that is, 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 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 $M$ 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 $d=1$ a complete classification is provided for them. Applications to orthonormal Haar systems in $\mathbb R^d$ and to integer univariate tiles are considered.
Bibliography: 18 titles.