Complete Characterization of Polyhedral Self-Affine Tiles

Abstract

A self-affine tile is a compact set Gsubset {mathbb R}^d that admits a partition (tiling) by parallel shifts of the set M^{-1}G, where M is an expanding matrix. We find all self-affine tiles which are polyhedral sets, i.e., unions of finitely many convex polyhedra. It is shown that there exists an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to integral self-affine tiles with standard digit sets, when the matrix M and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.