On the Characterization of Expansion Maps for Self-Affine Tilings
We consider self-affine tilings inRn with expansion matrixand address the question which matricescan arise this way. In one dimension, � is an expansion factor of a self-affine tiling if and only if|�| is a Perron number, by a result of Lind. In two dimensions, whenis a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complexis an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition forto be an expansion matrix for any n, assuming only thatis diagonalizable over C. We conjecture that this condition onis also sufficient for the existence of a self-affine tiling.