Higher rank graphs, k-subshifts and k-automata

Given a $k$-graph $\Lambda$ we construct a Markov space $M_\Lambda$, and a collection of $k$ pairwise commuting cellular automata on $M_\Lambda$, providing for a factorization of Markov's shift. Iterating these maps we obtain an action of ${\mathbb N}^k$ on $M_\Lambda$ which is then used to form a semidirect product groupoid $M_\Lambda \rtimes {\mathbb N}^k$. This groupoid turns out to be identical to the path groupoid constructed by Kumjian and Pask, and hence its C*-algebra is isomorphic to the higher rank graph C*-algebra of $\Lambda$.