Simplicity of 2-graph algebras associated to Dynamical Systems

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $\mathrm{\Lambda }$ we consider has an associated $C^{\ast }$-algebra, denoted $C^{\ast }(\mathrm{\Lambda })$, which is simple and purely infinite when $\mathrm{\Lambda }$ is aperiodic. We give new, straightforward conditions which ensure that $\mathrm{\Lambda }$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $\mathrm{\Lambda }$ in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.