Cocycles on Certain Groupoids Associated to $ \mathbb{N}^{k} $-Actions

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $C^{\ast}$-algebras associated to these groupoids. We provide a new characterization of $1$-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $k$-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($k$-Ruelle triples and commuting Ruelle operators). Results on KMS states on $C^{\ast}$-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.