Canonical self-affine tilings by iterated function systems Erin Pearse 28A8028A7552A0552A2052B9952C2052C2226B2549Q1551F9951M2051M2552A2252A3852A3952C4553C6554F45iterated function systemcomplex dimensionszeta functiontube formulafractal Steiner formulainradiusself-affineself-similartilingcurvature matrixgenerating function for the geometrydistributional explicit formulafractal string 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.) Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3220 10.1512/iumj.2007.56.3220 en Indiana Univ. Math. J. 56 (2007) 3151 - 3170 state-of-the-art mathematics http://iumj.org/access/