Canonical self-affine tilings by iterated function systems

Abstract

An iterated function system $\Phi$ consisting of contractive similarity mappings has a unique attractor $F \subseteq \mathbb{R}^d$ 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 $\mathcal{T}$ of the convex hull of the attractor. More precisely, it tiles the complement of the attractor within its convex hull. 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 $\Phi$. In particular, the tiles encode all the scaling data of $\Phi$. We give the construction, along with some examples and applications. The tiling $\mathcal{T}$ is the foundation for the higher-dimensional extension of the theory of \emph{complex dimensions} which was developed for the case $d=1$ in ``Fractal Geometry, Complex Dimensions, and Zeros of Zeta Functions,'' by Michel L. Lapidus and Machiel van Frankenhuijsen.