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 x 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 mu(T) > 0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this.