article

Title: Canonical self-affine tilings by iterated function systems

Authors: Erin P. J. Pearse

Issue: Volume 56 (2007), Issue 6, 3151-3170

Abstract:

$An iterated function system $\Phi$ consisting of contractive affine mappings has a unique attractor $F \subseteq \mathbb{R}^d$ which is invariant under the action of the system, as was shown by Hutchinson [J.E. Hutchinson, \textit{Fractals and self-similarity}, Indiana Univ. Math. J. \textbf{30} (1981), 713--747]. This paper shows how the action of the function system naturally produces a tiling $\mathcal{T}$ 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 $\Phi$. 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 in Lapidus-van Frankenhuijsen for the case $d=1$ (see M.L. Lapidus and M. van Frankenhuijsen, \textit{Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings}. New York: Springer-Verlag (Springer Monographs in Mathematics), 2006.)$