Higher rank graphs, k-subshifts and k-automata

TL;DR

This paper constructs a new dynamical system from higher rank graphs using Markov spaces and cellular automata, linking it to known C*-algebras of these graphs through groupoid isomorphisms.

Contribution

It introduces a novel approach to realize higher rank graph C*-algebras via Markov spaces and cellular automata, establishing a connection with Kumjian and Pask's path groupoid.

Findings

The semidirect product groupoid is isomorphic to the higher rank graph's path groupoid.

The constructed C*-algebra matches the higher rank graph C*-algebra.

Provides a dynamical systems perspective on higher rank graph algebras.

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