Abstract
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.