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.