Purely infinite simple $C^*$-algebras associated to integer dilation matrices Astrid an HuefRuy ExelIain Raeburn 46L55crossed productendomorphismtransfer operatorinteger dilation matrixpurely infinitesimple$K$-theory Given an $n \times n$ integer matrix $A$ whose eigenvalues are strictly greater than $1$ in absolute value, let $\sigma_A$ be the transformation of the $n$-torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ defined by $\sigma_A(e^{2\pi ix}) = e^{2\pi iAx}$ for $x \in \mathbb{R}^n$. We study the associated crossed-product $C^{*}$-algebra, which is defined using a certain transfer operator for $\sigma_A$, proving it to be simple and purely infinite and computing its $K$-theory groups. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4331 10.1512/iumj.2011.60.4331 en Indiana Univ. Math. J. 60 (2011) 1033 - 1058 state-of-the-art mathematics http://iumj.org/access/