Classification of Integral Expanding Matrices and Self-Affine Tiles

Let T be a self-affine tile that is generated by an expanding integral matrix A and a digit set D. It is known that many properties of T are invariant under the Z-similarity of the matrix A. In [LW1] Lagarias and Wang showed that if A is a 2 × 2 expanding matrix with |det(A)| = 2, then the Z-similar class is uniquely determined by the characteristic polynomial of A. This is not true if |det(A)| > 2. In this paper we give complete classifications of the Z-similar classes for the cases |det(A)| = 3, 4, 5. We then make use of the classification for |det(A)| = 3 to consider the digit set D of the tile and show that µ(T) > 0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this.