In this talk, I shall present an algebraic classification of all continuous $\mathbb{R}$-valued one-cocycles on the Deaconu-Renault groupoid associated to a compact metrizable space $X$ and a finite family $\sigma$ of commuting surjective local homeomorphisms on $X$. This classification result enables us to characterize, for every such cocycle $c$, the set of all probability regular Borel measures on $X$ that are quasi-invariant for $(X,\sigma)$ with Radon-Nikodym derivative $c$ — completely in terms of ergodic-theoretic objects known as Ruelle transfer operators.

I shall also present a generalized version of the Ruelle-Perron-Frobenius Theorem that says that if $(\sigma,c)$ satisfies some mild topological and metrical conditions, then a Ruelle-Perron-Frobenius eigenmeasure on $X$ exists, which allows us to construct KMS states on the corresponding groupoid $C^{\ast}$-algebra for the generalized gauge dynamics associated to $c$.

Some applications involving discrete and topological higher-rank graphs shall be shown. It is hoped that the ideas contained in this talk will appeal to $C^{\ast}$-algebraists and ergodic theorists alike.

This is joint work with Carla Farsi, Alex Kumjian, and Judith Packer.