Periodicity in Rank 2 Graph Algebras

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\text{ca}(\text{Fth})$. The periodic C*-algebras are characterized, and it is shown that $\text{ca}(\text{Fth})\simeq \text{rC}(\text{bT})\otimes \text{fA}$ where $\text{fA}$ is a simple C*-algebra.