On the Characterization of Expansion Maps for Self-Affine Tilings

We consider self-affine tilings in a"e (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 a",. We conjecture that this condition on phi is also sufficient for the existence of a self-affine tiling.