KMS States of Self-Similar $k$-Graph C*-Algebras

Let $G$ be a countable discrete amenable group, and $\Lambda$ be a strongly connected finite $k$-graph. If $(G,\Lambda)$ is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra $\O_{G,\Lambda}$ associated to $(G,\Lambda)$ is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group $\Per_{G,\Lambda}$ of $(G,\Lambda)$, depending on whether the Perron-Frobenius eigenvector of $\Lambda$ preserves the $G$-action. As applications of our main results, we also exhibit several classes of important examples.