Higher rank graphs, k-subshifts and k-automata

Abstract

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 $M_{\Lambda }⋊{\mathbb{N}}^k$. 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 $\Lambda$.