On the characterization of expansion maps for self-affine tilings

Abstract

We consider self-affine tilings in $\R^n$ with expansion matrix $\phi$ and address the question which matrices $\phi$ can arise this way. In one dimension, $\lambda$ is an expansion factor of a self-affine tiling if and only if $|\lambda|$ is a Perron number, by a result of Lind. In two dimensions, when $\phi$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $\lambda$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $\phi$ to be an expansion matrix for any $n$, assuming only that $\phi$ is diagonalizable over the complex numbers. We conjecture that this condition on $\phi$ is also sufficient for the existence of a self-affine tiling.