Classification of integral expanding matrices and self-affine tiles
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AbstractLet 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.
All Author(s) ListKirat I, Lau KS
Journal nameDiscrete and Computational Geometry
Volume Number28
Issue Number1
Pages49 - 73
LanguagesEnglish-United Kingdom
Web of Science Subject CategoriesComputer Science; Computer Science, Theory & Methods; COMPUTER SCIENCE, THEORY & METHODS; Mathematics; MATHEMATICS

Last updated on 2021-27-02 at 01:32