Canonical Self-similar Tilings by Ifs

An iterated function system consisting of contractive similarity mappings has a unique attractor F ⊆ Rd which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling T of the convex hull of the attractor. These tiles form a collection of sets whose geometry is typically much simpler than that of F , yet retains key information about both F and Φ. In particular, the tiles encode all the scaling data of Φ. We give the construction, along with some examples and applications. The tiling T is the foundation for the higher-dimensional extension of the theory of complex dimensions which was developed in [La-vF1] for the case d = 1.