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

An action of ${\mathbb{Z}}^k$ is associated to a higher rank graph $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ is strongly Morita equivalent to $C^{\ast }(\mathrm{\Lambda })$. Hence, if $\mathrm{\Lambda }$ satisfies the aperiodicity condition, the stable Ruelle algebra is simple, stable and purely infinite.