Classification of integral expanding matrices and self-affine tiles
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 their classification the authors further show that in the case \(|\det A|=3\), \(\mu(T)>0\) if and only if \(D\) is a standard digit set.
scientific article; zbMATH DE number 1784796