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

Abstract

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

Sixteen pages with no figures. To appear Ergodic Theory and Dynamical Systems