Classification of integral expanding matrices and self-affine tiles
type: Item
instance of: scholarly article
title: Classification of integral expanding matrices and self-affine tiles
zbMATH Open document ID: 1020.37003
author: İbrahim Kırat
author: Ka-Sing Lau
published in: Discrete \& Computational Geometry
publication date: 2003-10-09T00:00:00Z
review text: An expanding integral matrix is \(A\in M_n(\mathbb{Z})\) with all eigenvalues of absolute value \(>1\). If \(|\det A|=q\), let \(D=\{d_1, \dots ,d_q\} \subseteq \mathbb{Z}^n\) be a set of \(q\) distinct vectors, called a \(q\)-digit set. The affine maps \(w_j(x)=A^{-1}(x+d_j)\), \(j=1,\dots,q\), form a so-called iterated function system. The unique compact set \(T\) such that \(T= \bigcup^q_{j= 1} w_j(T)\) is called the attractor of the system and is explicitly given by \(T=T(A,D)= \sum^\infty_{i=1} A^{-i}d_{j_i}: d_{j_i}\in D\). Its Lebesgue measure \(\mu(T)\) is an integer; if \(\mu(T)>0\), \(T\) is called an integral self-affine tile. In order to classify expanding integral matrices it is useful to consider \(\mathbb{Z}\)-similarity of matrices: \(A\sim B\) if there exists a unimodular matrix \(P\in M_n(\mathbb{Z})\) such that \(P^{-1}AP=B\). Lagarias and Wang used characteristic polynomials to show that each integral expanding polynomial (with all roots of absolute value \(>1)\) \(f(x)= x^2 +ax+q\), \(|q|=2\), corresponds to exactly one \(\mathbb{Z}\)-similar class of expanding matrices \(A\in M_2 (\mathbb{Z})\), and since there are 6 such polynomials, there are exactly 6 \(\mathbb{Z}\)-similar classes of \(2\times 2\) expanding integral matrices \(A\) with \(|\det A |=2\). The authors consider the classification problem of \(A\in M_2 (\mathbb{Z})\) with \(|\det A|=3\), 4 and 5. For \(|\det A|=3\) there are 10 characteristic polynomials; for 9 of them \(A\) is \(\mathbb{Z}\)-similar to the companion matrix. But for \(f(x)=x^2+3\), \(A\) is \(\mathbb{Z}\)-similar to either the companion matrix or to \(\left(\begin{smallmatrix} 1 & 2\\ -2 & -1\end{smallmatrix} \right)\). For \(|\det A|=4\) there are 14 characteristic polynomials, 4 with 1, 7 with 2, 1 with 3, and 2 with \(\infty\) many \(\mathbb{Z}\)-similarity classes, respectively. For \(|\det A|=5\) there are 18 characteristic polynomials, 14 with unique, and 4 with two \(\mathbb{Z}\)-similarity classes. Using …
Mathematics Subject Classification ID: 37B50
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1784796
zbMATH Keywords: self-affine tiles
zbMATH Keywords: \(\mathbb{Z}\)-similarity of matrices
zbMATH Keywords: iterated function system
zbMATH Keywords: attractor
zbMATH Keywords: expanding integral matrices
zbMATH Keywords: characteristic polynomials
reviewed by: Jonathan Schaer
MaRDI profile type: Publication
full work available at URL: s00454-001-0091-2
OpenAlex ID: W2024190285
DOI: 10.1007/S00454-001-0091-2
description: scientific article; zbMATH DE number 1784796