Actions of ${\mathbb Z}^k$ associated to higher rnak graphs

Abstract

An action of ${\mathbb Z}^k$ is associated to a higher rank graph $Λ$ satisfying a mild assumption. This generalises the construction of a topological Markov shift arising from a nonnegative integer matrix. We show that the stable Ruelle algebra of $Λ$ is strongly Morita equivalent to $C^*(Λ)$. Hence, if $Λ$ satisfies the aperiodicity condition, the stable Ruelle algebra is simple, stable and purely infinite.