On the characterization of expansion maps for self-affine tilings

Abstract

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

Revised version. A typo corrected (after publication!) in the definition of the set $Ω$ at the bottom of p.13