Discrete & Computational Geometry

, Volume 28, Issue 1, pp 49–73

Classification of Integral Expanding Matrices and Self-Affine Tiles

Article

DOI: 10.1007/s00454-001-0091-2

Cite this article as:
Kirat & Lau Discrete Comput Geom (2002) 28: 49. doi:10.1007/s00454-001-0091-2

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.

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Kirat
    • 1
  • Lau
    • 2
  1. 1.Department of Mathematics, Sakarya University, Sakarya, Turkey ibkst@yahoo.comTurkey
  2. 2.Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong kslau@math.cuhk.edu.hkHK