Asymptotic C ∗ -Algebras from ℤ k -Actions on Higher Rank Graphs

Abstract

For a dynamical system arising from ${\mathbb{Z}}^k$-action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the $k$-graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.