Cocycles on groupoids arising from -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 and uniqueness results for KMS states associated to the graphs.