Abstract
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.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kirat, Lau Classification of Integral Expanding Matrices and Self-Affine Tiles. Discrete Comput Geom 28, 49–73 (2002). https://doi.org/10.1007/s00454-001-0091-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-001-0091-2